Physics Book - User contributions [en]

Velocity

2021-11-29T02:12:55Z

Vkumar350: /* Derivative Relationships */

'''CLAIMED BY VIBHAV KUMAR (FALL 2021)'''

==Main Idea==

Velocity is a [[Vectors|vector]] quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol <math>\vec{v}</math> or v, as opposed to <math>v</math>, which denotes [[Speed|speed]]. Velocity is defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as [[Linear Momentum|momentum]] and [[Magnetic Force|magnetic force]], are functions of velocity. Velocity is given in unit distance per unit time. The [[SI Units|SI unit]] for velocity is the meter per second (m/s).

===A Mathematical Model===

Instantaneous velocity <math>\vec{v}</math> is defined as:

<math>\vec{v} = \frac{d\vec{r}}{dt}</math>

where <math>\vec{r}</math> is a position vector and <math>t</math> is time.

====Average Velocity====

Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted <math>\vec{v}_{avg}</math>. The average velocity over an interval of time <math>\Delta t</math> is given by

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

where <math>\Delta \vec{r}</math> is the displacement (change in position) over that time interval (<math>\Delta \vec{r} = \vec{r}_f - \vec{r}_i</math>).

Average velocity is often confused with average speed. A section on the [[Speed vs Velocity]] page goes into depth about the difference.

As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity <math>\vec{v}_i</math> and final velocity <math>\vec{v}_f</math>:

<math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math>

Unlike the first equation, this equation is only true if [[Acceleration|acceleration]] is constant.

====Instantaneous Velocity====

Instantaneous velocity is defined as the velocity of an object in motion at a specific point in time. Determining instantaneous velocity is similar to determining the average velocity. However, in instantaneous velocity, we shorten the period of time over which we measure the change in distance to a point where the time has approached zero. While instantaneous velocity is defined as the velocity of a body at a specific point in time, the average velocity is the displacement over the time taken for the body to displace itself. Visually, this can be seen as the following:

[[File:Instantaneous_velocity.PNG|200px|thumb|left|Graph Displaying Relationship Between Instantaneous and Average Velocity]]

The instantaneous velocity can be found by using calculus and taking the derivate of a distance vs time function at a specific point in time. If the position function is ''f(t)'', then the velocity at a time t would be ''v(t) = f'(t)''.

====Velocity and Acceleration Relationship====

Similar to how the velocity is defined as a change in distance over time, acceleration can be defined as the change in velocity over time. Acceleration can be calculated by taking the derivative of velocity with respect to time. If the velocity function was ''v(t)'', then the acceleration at time t would be ''a(t) = v'(t) = f"(t)''. A constant non-zero acceleration means that the velocity is increasing or decreasing linearly. If the acceleration is always 0 and velocity is non-zero, this means that velocity is constant and position is increasing or decreasing linearly.

====Derivative Relationships====

Velocity is the time derivative of position:

<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.

[[Acceleration]], in turn, is the time derivative of velocity:

<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>

====Integral Relationships====

Position is the time integral of velocity:

<math>\vec{r}(t) = \int \vec{v}(t) \ dt</math>.

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

<math>\Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \vec{v}(t) \ dt</math>.

Velocity is, in turn, the time integral of acceleration:

<math>\vec{v}(t) = \int \vec{a}(t) \ dt</math>.

====Kinematic Equations====

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the [[Kinematics]] page.

====In Physics====

According to [[Newton's First Law of Motion]], the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. [[Newton's Second Law: the Momentum Principle]] describes how velocity changes over time as a result of forces.

===A Computational Model===

In VPython simulations of physical systems, the [[Iterative Prediction]] algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]

Click "view this program" in the top left corner to view the source code.

==Examples==

===Simple===

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

<math>\vec{v}_{avg} = \frac{<-2, 8> - <4,-1>}{3}</math>

<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>

<math>\vec{v}_{avg} = <-2, 3></math>m/s

therefore

<math>\vec{v}(t=2) = <-2, 3></math>m/s

===Middling===

A particle's position as a function of time is as follows: <math>\vec{r}(t) = <6\sin(2 \pi t), -6t^3 + 10, e^{8t}></math>

What is the particle's velocity as a function of time?

Solution:

<math>\vec{v} = \frac{d \vec{r}}{dt}</math>

<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>

===Difficult===

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity <math>v</math> varies over time, obeying the following function:

<math>
v(t) =
\begin{cases}
4-2t & \text{if $t\leq 3$} \\
-2 & \text{if $t>3$}
\end{cases}
</math>m/s

What is its position as a function of time after time t=0?

Solution:

<math>x(t) = x(0) + \int_0^t v(t')dt'</math> (<math>t'</math> is a "dummy variable" since <math>t</math> is already our limit of integration)

for <math>t\leq 3</math>,

<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>

<math>x(t) = 2 + [4t'-t'^2]_0^t</math>

<math>x(t) = 2 + 4t - t^2</math>

for <math>t>3</math>,

<math>x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt'</math>

<math>x(t) = 5 + \int_3^t -2 dt'</math>

<math>x(t) = 5 + [-2t']_3^t</math>

<math>x(t) = 5 -2t + 6</math>

<math>x(t) = 11 - 2t</math>

final answer:

<math>
x(t) =
\begin{cases}
-t^2 + 4t + 2 & \text{if $t\leq 3$} \\
11-2t & \text{if $t>3$}
\end{cases}
</math>m

==Connectedness==

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Linear Momentum]], [[Total Angular Momentum|Angular Momentum]], and [[Magnetic Force]] depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

==See Also==

*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]

===External links===

[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]

[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average Velocity]

[https://www.youtube.com/watch?v=Bxp0AWhs57g YouTube video explaining average vs instantaneous velocity]

==References==

1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.

[[Category:Properties of Matter]]

Velocity

2021-11-29T02:12:46Z

Vkumar350: /* A Mathematical Model */

Velocity

2021-11-29T02:12:09Z

Vkumar350:

Velocity

2021-11-29T02:02:45Z

Vkumar350:

'''CLAIMED BY VIBHAV KUMAR (FALL 2021)'''

==Main Idea==

Velocity is a [[Vectors|vector]] quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol <math>\vec{v}</math> or v, as opposed to <math>v</math>, which denotes [[Speed|speed]]. Velocity is defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as [[Linear Momentum|momentum]] and [[Magnetic Force|magnetic force]], are functions of velocity. Velocity is given in unit distance per unit time. The [[SI Units|SI unit]] for velocity is the meter per second (m/s).

===A Mathematical Model===

Instantaneous velocity <math>\vec{v}</math> is defined as:

<math>\vec{v} = \frac{d\vec{r}}{dt}</math>

where <math>\vec{r}</math> is a position vector and <math>t</math> is time.

====Average Velocity====

Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted <math>\vec{v}_{avg}</math>. The average velocity over an interval of time <math>\Delta t</math> is given by

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

where <math>\Delta \vec{r}</math> is the displacement (change in position) over that time interval (<math>\Delta \vec{r} = \vec{r}_f - \vec{r}_i</math>).

Average velocity is often confused with average speed. A section on the [[Speed vs Velocity]] page goes into depth about the difference.

As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity <math>\vec{v}_i</math> and final velocity <math>\vec{v}_f</math>:

<math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math>

Unlike the first equation, this equation is only true if [[Acceleration|acceleration]] is constant.

====Instantaneous Velocity====

Instantaneous velocity is defined as the velocity of an object in motion at a specific point in time. Determining instantaneous velocity is similar to determining the average velocity. However, in instantaneous velocity, we shorten the period of time over which we measure the change in distance to a point where the time has approached zero. While instantaneous velocity is defined as the velocity of a body at a specific point in time, the average velocity is the displacement over the time taken for the body to displace itself. Visually, this can be seen as the following:

[[File:Instantaneous_velocity.PNG|200px|thumb|left|Graph Displaying Relationship Between Instantaneous and Average Velocity]]

The instantaneous velocity can be found by using calculus and taking the derivate of a distance vs time function at a specific point in time. If the position function is ''f(t)'', then the velocity at a time t would be ''v(t) = f'(t)''.

====Derivative Relationships====

Velocity is the time derivative of position:

<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.

[[Acceleration]], in turn, is the time derivative of velocity:

<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>

====Integral Relationships====

Position is the time integral of velocity:

<math>\vec{r}(t) = \int \vec{v}(t) \ dt</math>.

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

<math>\Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \vec{v}(t) \ dt</math>.

Velocity is, in turn, the time integral of acceleration:

<math>\vec{v}(t) = \int \vec{a}(t) \ dt</math>.

====Kinematic Equations====

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the [[Kinematics]] page.

====In Physics====

According to [[Newton's First Law of Motion]], the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. [[Newton's Second Law: the Momentum Principle]] describes how velocity changes over time as a result of forces.

===A Computational Model===

In VPython simulations of physical systems, the [[Iterative Prediction]] algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]

Click "view this program" in the top left corner to view the source code.

==Examples==

===Simple===

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

<math>\vec{v}_{avg} = \frac{<-2, 8> - <4,-1>}{3}</math>

<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>

<math>\vec{v}_{avg} = <-2, 3></math>m/s

therefore

<math>\vec{v}(t=2) = <-2, 3></math>m/s

===Middling===

A particle's position as a function of time is as follows: <math>\vec{r}(t) = <6\sin(2 \pi t), -6t^3 + 10, e^{8t}></math>

What is the particle's velocity as a function of time?

Solution:

<math>\vec{v} = \frac{d \vec{r}}{dt}</math>

<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>

===Difficult===

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity <math>v</math> varies over time, obeying the following function:

<math>
v(t) =
\begin{cases}
4-2t & \text{if $t\leq 3$} \\
-2 & \text{if $t>3$}
\end{cases}
</math>m/s

What is its position as a function of time after time t=0?

Solution:

<math>x(t) = x(0) + \int_0^t v(t')dt'</math> (<math>t'</math> is a "dummy variable" since <math>t</math> is already our limit of integration)

for <math>t\leq 3</math>,

<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>

<math>x(t) = 2 + [4t'-t'^2]_0^t</math>

<math>x(t) = 2 + 4t - t^2</math>

for <math>t>3</math>,

<math>x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt'</math>

<math>x(t) = 5 + \int_3^t -2 dt'</math>

<math>x(t) = 5 + [-2t']_3^t</math>

<math>x(t) = 5 -2t + 6</math>

<math>x(t) = 11 - 2t</math>

final answer:

<math>
x(t) =
\begin{cases}
-t^2 + 4t + 2 & \text{if $t\leq 3$} \\
11-2t & \text{if $t>3$}
\end{cases}
</math>m

==Connectedness==

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Linear Momentum]], [[Total Angular Momentum|Angular Momentum]], and [[Magnetic Force]] depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

==See Also==

*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]

===External links===

[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]

[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average Velocity]

[https://www.youtube.com/watch?v=Bxp0AWhs57g YouTube video explaining average vs instantaneous velocity]

==References==

1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.

[[Category:Properties of Matter]]

Velocity

2021-11-29T02:00:43Z

Vkumar350: /* A Mathematical Model */

'''CLAIMED BY VIBHAV KUMAR (FALL 2021)'''

from 2019 - NEEDS EDITING TO MOTIVATE THE CONCEPT OF VELOCITY FOR NOVICES: HERE'S ONE WAY TO DO THIS...INSERT DISCUSSION THAT INDICATES HOW ONE TAKES OBSERVATIONS OF THE POSITION OF AN OBJECT AT DIFFERENT TIMES AND USES THAT INFORMATION TO OBTAIN A VECTOR QUANTITY WHOSE MAGNITUDE TELLS ONE HOW FAST THE OBJECT IS MOVING AND WHOSE DIRECTION INDICATES THE DIRECTION THAT THE OBJECT IS MOVING.'''This page defines and describes velocity. Be sure to distinguish between velocity and [[Speed|speed]]. See [[Speed vs Velocity]] for similarities and differences.

==Main Idea==

Velocity is a [[Vectors|vector]] quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol <math>\vec{v}</math> or v, as opposed to <math>v</math>, which denotes [[Speed|speed]]. Velocity is defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as [[Linear Momentum|momentum]] and [[Magnetic Force|magnetic force]], are functions of velocity. Velocity is given in unit distance per unit time. The [[SI Units|SI unit]] for velocity is the meter per second (m/s).

===A Mathematical Model===

Instantaneous velocity <math>\vec{v}</math> is defined as:

<math>\vec{v} = \frac{d\vec{r}}{dt}</math>

where <math>\vec{r}</math> is a position vector and <math>t</math> is time.

====Average Velocity====

Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted <math>\vec{v}_{avg}</math>. The average velocity over an interval of time <math>\Delta t</math> is given by

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

where <math>\Delta \vec{r}</math> is the displacement (change in position) over that time interval (<math>\Delta \vec{r} = \vec{r}_f - \vec{r}_i</math>).

Average velocity is often confused with average speed. A section on the [[Speed vs Velocity]] page goes into depth about the difference.

As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity <math>\vec{v}_i</math> and final velocity <math>\vec{v}_f</math>:

<math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math>

Unlike the first equation, this equation is only true if [[Acceleration|acceleration]] is constant.

====Instantaneous Velocity====

Instantaneous velocity is defined as the velocity of an object in motion at a specific point in time. Determining instantaneous velocity is similar to determining the average velocity. However, in instantaneous velocity, we shorten the period of time over which we measure the change in distance to a point where the time has approached zero. While instantaneous velocity is defined as the velocity of a body at a specific point in time, the average velocity is the displacement over the time taken for the body to displace itself. Visually, this can be seen as the following:

[[File:Instantaneous_velocity.PNG|200px|thumb|left|Graph Displaying Relationship Between Instantaneous and Average Velocity]]

The instantaneous velocity can be found by using calculus and taking the derivate of a distance vs time function at a specific point in time. If the position function is ''f(t)'', then the velocity at a time t would be ''v(t) = f'(t)''.

====Derivative Relationships====

Velocity is the time derivative of position:

<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.

[[Acceleration]], in turn, is the time derivative of velocity:

<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>

====Integral Relationships====

Position is the time integral of velocity:

<math>\vec{r}(t) = \int \vec{v}(t) \ dt</math>.

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

<math>\Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \vec{v}(t) \ dt</math>.

Velocity is, in turn, the time integral of acceleration:

<math>\vec{v}(t) = \int \vec{a}(t) \ dt</math>.

====Kinematic Equations====

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the [[Kinematics]] page.

====In Physics====

According to [[Newton's First Law of Motion]], the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. [[Newton's Second Law: the Momentum Principle]] describes how velocity changes over time as a result of forces.

===A Computational Model===

In VPython simulations of physical systems, the [[Iterative Prediction]] algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]

Click "view this program" in the top left corner to view the source code.

==Examples==

===Simple===

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

<math>\vec{v}_{avg} = \frac{<-2, 8> - <4,-1>}{3}</math>

<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>

<math>\vec{v}_{avg} = <-2, 3></math>m/s

therefore

<math>\vec{v}(t=2) = <-2, 3></math>m/s

===Middling===

A particle's position as a function of time is as follows: <math>\vec{r}(t) = <6\sin(2 \pi t), -6t^3 + 10, e^{8t}></math>

What is the particle's velocity as a function of time?

Solution:

<math>\vec{v} = \frac{d \vec{r}}{dt}</math>

<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>

===Difficult===

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity <math>v</math> varies over time, obeying the following function:

<math>
v(t) =
\begin{cases}
4-2t & \text{if $t\leq 3$} \\
-2 & \text{if $t>3$}
\end{cases}
</math>m/s

What is its position as a function of time after time t=0?

Solution:

<math>x(t) = x(0) + \int_0^t v(t')dt'</math> (<math>t'</math> is a "dummy variable" since <math>t</math> is already our limit of integration)

for <math>t\leq 3</math>,

<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>

<math>x(t) = 2 + [4t'-t'^2]_0^t</math>

<math>x(t) = 2 + 4t - t^2</math>

for <math>t>3</math>,

<math>x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt'</math>

<math>x(t) = 5 + \int_3^t -2 dt'</math>

<math>x(t) = 5 + [-2t']_3^t</math>

<math>x(t) = 5 -2t + 6</math>

<math>x(t) = 11 - 2t</math>

final answer:

<math>
x(t) =
\begin{cases}
-t^2 + 4t + 2 & \text{if $t\leq 3$} \\
11-2t & \text{if $t>3$}
\end{cases}
</math>m

==Connectedness==

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Linear Momentum]], [[Total Angular Momentum|Angular Momentum]], and [[Magnetic Force]] depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

==See Also==

*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]

===External links===

[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]

[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average Velocity]

[https://www.youtube.com/watch?v=Bxp0AWhs57g YouTube video explaining average vs instantaneous velocity]

==References==

1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.

[[Category:Properties of Matter]]

Velocity

2021-11-29T01:59:55Z

Vkumar350: /* Derivative Relationships */

Velocity

2021-11-29T01:59:36Z

Vkumar350:

File:Instantaneous velocity.PNG

2021-11-29T01:46:58Z

Vkumar350:

Velocity

2021-11-29T01:07:19Z

Vkumar350: /* A Mathematical Model */

'''CLAIMED BY VIBHAV KUMAR (FALL 2021)'''

from 2019 - NEEDS EDITING TO MOTIVATE THE CONCEPT OF VELOCITY FOR NOVICES: HERE'S ONE WAY TO DO THIS...INSERT DISCUSSION THAT INDICATES HOW ONE TAKES OBSERVATIONS OF THE POSITION OF AN OBJECT AT DIFFERENT TIMES AND USES THAT INFORMATION TO OBTAIN A VECTOR QUANTITY WHOSE MAGNITUDE TELLS ONE HOW FAST THE OBJECT IS MOVING AND WHOSE DIRECTION INDICATES THE DIRECTION THAT THE OBJECT IS MOVING.'''This page defines and describes velocity. Be sure to distinguish between velocity and [[Speed|speed]]. See [[Speed vs Velocity]] for similarities and differences.

==Main Idea==

Velocity is a [[Vectors|vector]] quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol <math>\vec{v}</math> or v, as opposed to <math>v</math>, which denotes [[Speed|speed]]. Velocity is defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as [[Linear Momentum|momentum]] and [[Magnetic Force|magnetic force]], are functions of velocity. Velocity is given in unit distance per unit time. The [[SI Units|SI unit]] for velocity is the meter per second (m/s).

===A Mathematical Model===

Instantaneous velocity <math>\vec{v}</math> is defined as:

<math>\vec{v} = \frac{d\vec{r}}{dt}</math>

where <math>\vec{r}</math> is a position vector and <math>t</math> is time.

====Average Velocity====

Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted <math>\vec{v}_{avg}</math>. The average velocity over an interval of time <math>\Delta t</math> is given by

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

where <math>\Delta \vec{r}</math> is the displacement (change in position) over that time interval (<math>\Delta \vec{r} = \vec{r}_f - \vec{r}_i</math>).

Average velocity is often confused with average speed. A section on the [[Speed vs Velocity]] page goes into depth about the difference.

As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity <math>\vec{v}_i</math> and final velocity <math>\vec{v}_f</math>:

<math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math>

Unlike the first equation, this equation is only true if [[Acceleration|acceleration]] is constant.

====Instantaneous Velocity====

Instantaneous velocity is defined as the velocity of an object in motion at a specific point in time. Determining instantaneous velocity is similar to determining the average velocity. However, in instantaneous velocity, we shorten the period of time over which we measure the change in distance to a point where the time has approached zero. While instantaneous velocity is defined as the velocity of a body at a specific point in time, the average velocity is the displacement over the time taken for the body to displace itself. Visually, this can be seen as the following:
.

====Derivative Relationships====

Velocity is the time derivative of position:

<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.

[[Acceleration]], in turn, is the time derivative of velocity:

<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>

====Integral Relationships====

Position is the time integral of velocity:

<math>\vec{r}(t) = \int \vec{v}(t) \ dt</math>.

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

<math>\Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \vec{v}(t) \ dt</math>.

Velocity is, in turn, the time integral of acceleration:

<math>\vec{v}(t) = \int \vec{a}(t) \ dt</math>.

====Kinematic Equations====

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the [[Kinematics]] page.

====In Physics====

According to [[Newton's First Law of Motion]], the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. [[Newton's Second Law: the Momentum Principle]] describes how velocity changes over time as a result of forces.

===A Computational Model===

In VPython simulations of physical systems, the [[Iterative Prediction]] algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]

Click "view this program" in the top left corner to view the source code.

==Examples==

===Simple===

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

<math>\vec{v}_{avg} = \frac{<-2, 8> - <4,-1>}{3}</math>

<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>

<math>\vec{v}_{avg} = <-2, 3></math>m/s

therefore

<math>\vec{v}(t=2) = <-2, 3></math>m/s

===Middling===

A particle's position as a function of time is as follows: <math>\vec{r}(t) = <6\sin(2 \pi t), -6t^3 + 10, e^{8t}></math>

What is the particle's velocity as a function of time?

Solution:

<math>\vec{v} = \frac{d \vec{r}}{dt}</math>

<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>

===Difficult===

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity <math>v</math> varies over time, obeying the following function:

<math>
v(t) =
\begin{cases}
4-2t & \text{if $t\leq 3$} \\
-2 & \text{if $t>3$}
\end{cases}
</math>m/s

What is its position as a function of time after time t=0?

Solution:

<math>x(t) = x(0) + \int_0^t v(t')dt'</math> (<math>t'</math> is a "dummy variable" since <math>t</math> is already our limit of integration)

for <math>t\leq 3</math>,

<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>

<math>x(t) = 2 + [4t'-t'^2]_0^t</math>

<math>x(t) = 2 + 4t - t^2</math>

for <math>t>3</math>,

<math>x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt'</math>

<math>x(t) = 5 + \int_3^t -2 dt'</math>

<math>x(t) = 5 + [-2t']_3^t</math>

<math>x(t) = 5 -2t + 6</math>

<math>x(t) = 11 - 2t</math>

final answer:

<math>
x(t) =
\begin{cases}
-t^2 + 4t + 2 & \text{if $t\leq 3$} \\
11-2t & \text{if $t>3$}
\end{cases}
</math>m

==Connectedness==

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Linear Momentum]], [[Total Angular Momentum|Angular Momentum]], and [[Magnetic Force]] depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

==See Also==

*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]

===External links===

[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]

[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average Velocity]

[https://www.youtube.com/watch?v=Bxp0AWhs57g YouTube video explaining average vs instantaneous velocity]

==References==

1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.

[[Category:Properties of Matter]]

Velocity

2021-11-29T01:00:47Z

Vkumar350: /* A Mathematical Model */

'''CLAIMED BY VIBHAV KUMAR (FALL 2021)'''

from 2019 - NEEDS EDITING TO MOTIVATE THE CONCEPT OF VELOCITY FOR NOVICES: HERE'S ONE WAY TO DO THIS...INSERT DISCUSSION THAT INDICATES HOW ONE TAKES OBSERVATIONS OF THE POSITION OF AN OBJECT AT DIFFERENT TIMES AND USES THAT INFORMATION TO OBTAIN A VECTOR QUANTITY WHOSE MAGNITUDE TELLS ONE HOW FAST THE OBJECT IS MOVING AND WHOSE DIRECTION INDICATES THE DIRECTION THAT THE OBJECT IS MOVING.'''This page defines and describes velocity. Be sure to distinguish between velocity and [[Speed|speed]]. See [[Speed vs Velocity]] for similarities and differences.

==Main Idea==

Velocity is a [[Vectors|vector]] quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol <math>\vec{v}</math> or v, as opposed to <math>v</math>, which denotes [[Speed|speed]]. Velocity is defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as [[Linear Momentum|momentum]] and [[Magnetic Force|magnetic force]], are functions of velocity. Velocity is given in unit distance per unit time. The [[SI Units|SI unit]] for velocity is the meter per second (m/s).

===A Mathematical Model===

Instantaneous velocity <math>\vec{v}</math> is defined as:

<math>\vec{v} = \frac{d\vec{r}}{dt}</math>

where <math>\vec{r}</math> is a position vector and <math>t</math> is time.

====Average Velocity====

Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted <math>\vec{v}_{avg}</math>. The average velocity over an interval of time <math>\Delta t</math> is given by

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

where <math>\Delta \vec{r}</math> is the displacement (change in position) over that time interval (<math>\Delta \vec{r} = \vec{r}_f - \vec{r}_i</math>).

Average velocity is often confused with average speed. A section on the [[Speed vs Velocity]] page goes into depth about the difference.

As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity <math>\vec{v}_i</math> and final velocity <math>\vec{v}_f</math>:

<math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math>

Unlike the first equation, this equation is only true if [[Acceleration|acceleration]] is constant.

====Instantaneous Velocity====

====Derivative Relationships====

Velocity is the time derivative of position:

<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.

[[Acceleration]], in turn, is the time derivative of velocity:

<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>

====Integral Relationships====

Position is the time integral of velocity:

<math>\vec{r}(t) = \int \vec{v}(t) \ dt</math>.

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

<math>\Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \vec{v}(t) \ dt</math>.

Velocity is, in turn, the time integral of acceleration:

<math>\vec{v}(t) = \int \vec{a}(t) \ dt</math>.

====Kinematic Equations====

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the [[Kinematics]] page.

====In Physics====

According to [[Newton's First Law of Motion]], the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. [[Newton's Second Law: the Momentum Principle]] describes how velocity changes over time as a result of forces.

===A Computational Model===

In VPython simulations of physical systems, the [[Iterative Prediction]] algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]

Click "view this program" in the top left corner to view the source code.

==Examples==

===Simple===

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

<math>\vec{v}_{avg} = \frac{<-2, 8> - <4,-1>}{3}</math>

<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>

<math>\vec{v}_{avg} = <-2, 3></math>m/s

therefore

<math>\vec{v}(t=2) = <-2, 3></math>m/s

===Middling===

A particle's position as a function of time is as follows: <math>\vec{r}(t) = <6\sin(2 \pi t), -6t^3 + 10, e^{8t}></math>

What is the particle's velocity as a function of time?

Solution:

<math>\vec{v} = \frac{d \vec{r}}{dt}</math>

<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>

===Difficult===

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity <math>v</math> varies over time, obeying the following function:

<math>
v(t) =
\begin{cases}
4-2t & \text{if $t\leq 3$} \\
-2 & \text{if $t>3$}
\end{cases}
</math>m/s

What is its position as a function of time after time t=0?

Solution:

<math>x(t) = x(0) + \int_0^t v(t')dt'</math> (<math>t'</math> is a "dummy variable" since <math>t</math> is already our limit of integration)

for <math>t\leq 3</math>,

<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>

<math>x(t) = 2 + [4t'-t'^2]_0^t</math>

<math>x(t) = 2 + 4t - t^2</math>

for <math>t>3</math>,

<math>x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt'</math>

<math>x(t) = 5 + \int_3^t -2 dt'</math>

<math>x(t) = 5 + [-2t']_3^t</math>

<math>x(t) = 5 -2t + 6</math>

<math>x(t) = 11 - 2t</math>

final answer:

<math>
x(t) =
\begin{cases}
-t^2 + 4t + 2 & \text{if $t\leq 3$} \\
11-2t & \text{if $t>3$}
\end{cases}
</math>m

==Connectedness==

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Linear Momentum]], [[Total Angular Momentum|Angular Momentum]], and [[Magnetic Force]] depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

==See Also==

*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]

===External links===

[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]

[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average Velocity]

[https://www.youtube.com/watch?v=Bxp0AWhs57g YouTube video explaining average vs instantaneous velocity]

==References==

1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.

[[Category:Properties of Matter]]

Velocity

2021-11-28T14:00:14Z

Vkumar350:

'''CLAIMED BY VIBHAV KUMAR (FALL 2021)'''

from 2019 - NEEDS EDITING TO MOTIVATE THE CONCEPT OF VELOCITY FOR NOVICES: HERE'S ONE WAY TO DO THIS...INSERT DISCUSSION THAT INDICATES HOW ONE TAKES OBSERVATIONS OF THE POSITION OF AN OBJECT AT DIFFERENT TIMES AND USES THAT INFORMATION TO OBTAIN A VECTOR QUANTITY WHOSE MAGNITUDE TELLS ONE HOW FAST THE OBJECT IS MOVING AND WHOSE DIRECTION INDICATES THE DIRECTION THAT THE OBJECT IS MOVING.'''This page defines and describes velocity. Be sure to distinguish between velocity and [[Speed|speed]]. See [[Speed vs Velocity]] for similarities and differences.

==Main Idea==

Velocity is a [[Vectors|vector]] quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol <math>\vec{v}</math> or v, as opposed to <math>v</math>, which denotes [[Speed|speed]]. Velocity is defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as [[Linear Momentum|momentum]] and [[Magnetic Force|magnetic force]], are functions of velocity. Velocity is given in unit distance per unit time. The [[SI Units|SI unit]] for velocity is the meter per second (m/s).

===A Mathematical Model===

Instantaneous velocity <math>\vec{v}</math> is defined as:

<math>\vec{v} = \frac{d\vec{r}}{dt}</math>

where <math>\vec{r}</math> is a position vector and <math>t</math> is time.

====Average Velocity====

Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted <math>\vec{v}_{avg}</math>. The average velocity over an interval of time <math>\Delta t</math> is given by

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

where <math>\Delta \vec{r}</math> is the displacement (change in position) over that time interval (<math>\Delta \vec{r} = \vec{r}_f - \vec{r}_i</math>).

Average velocity is often confused with average speed. A section on the [[Speed vs Velocity]] page goes into depth about the difference.

As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity <math>\vec{v}_i</math> and final velocity <math>\vec{v}_f</math>:

<math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math>

Unlike the first equation, this equation is only true if [[Acceleration|acceleration]] is constant.

====Derivative Relationships====

Velocity is the time derivative of position:

<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.

[[Acceleration]], in turn, is the time derivative of velocity:

<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>

====Integral Relationships====

Position is the time integral of velocity:

<math>\vec{r}(t) = \int \vec{v}(t) \ dt</math>.

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

<math>\Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \vec{v}(t) \ dt</math>.

Velocity is, in turn, the time integral of acceleration:

<math>\vec{v}(t) = \int \vec{a}(t) \ dt</math>.

====Kinematic Equations====

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the [[Kinematics]] page.

====In Physics====

According to [[Newton's First Law of Motion]], the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. [[Newton's Second Law: the Momentum Principle]] describes how velocity changes over time as a result of forces.

===A Computational Model===

In VPython simulations of physical systems, the [[Iterative Prediction]] algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]

Click "view this program" in the top left corner to view the source code.

==Examples==

===Simple===

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

<math>\vec{v}_{avg} = \frac{<-2, 8> - <4,-1>}{3}</math>

<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>

<math>\vec{v}_{avg} = <-2, 3></math>m/s

therefore

<math>\vec{v}(t=2) = <-2, 3></math>m/s

===Middling===

A particle's position as a function of time is as follows: <math>\vec{r}(t) = <6\sin(2 \pi t), -6t^3 + 10, e^{8t}></math>

What is the particle's velocity as a function of time?

Solution:

<math>\vec{v} = \frac{d \vec{r}}{dt}</math>

<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>

===Difficult===

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity <math>v</math> varies over time, obeying the following function:

<math>
v(t) =
\begin{cases}
4-2t & \text{if $t\leq 3$} \\
-2 & \text{if $t>3$}
\end{cases}
</math>m/s

What is its position as a function of time after time t=0?

Solution:

<math>x(t) = x(0) + \int_0^t v(t')dt'</math> (<math>t'</math> is a "dummy variable" since <math>t</math> is already our limit of integration)

for <math>t\leq 3</math>,

<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>

<math>x(t) = 2 + [4t'-t'^2]_0^t</math>

<math>x(t) = 2 + 4t - t^2</math>

for <math>t>3</math>,

<math>x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt'</math>

<math>x(t) = 5 + \int_3^t -2 dt'</math>

<math>x(t) = 5 + [-2t']_3^t</math>

<math>x(t) = 5 -2t + 6</math>

<math>x(t) = 11 - 2t</math>

final answer:

<math>
x(t) =
\begin{cases}
-t^2 + 4t + 2 & \text{if $t\leq 3$} \\
11-2t & \text{if $t>3$}
\end{cases}
</math>m

==Connectedness==

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Linear Momentum]], [[Total Angular Momentum|Angular Momentum]], and [[Magnetic Force]] depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

==See Also==

*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]

===External links===

[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]

[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average Velocity]

[https://www.youtube.com/watch?v=Bxp0AWhs57g YouTube video explaining average vs instantaneous velocity]

==References==

1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.

[[Category:Properties of Matter]]

Velocity

2021-11-28T13:58:07Z

Vkumar350:

CLAIMED BY VIBHAV KUMAR (FALL 2021)

from 2019 - NEEDS EDITING TO MOTIVATE THE CONCEPT OF VELOCITY FOR NOVICES: HERE'S ONE WAY TO DO THIS...INSERT DISCUSSION THAT INDICATES HOW ONE TAKES OBSERVATIONS OF THE POSITION OF AN OBJECT AT DIFFERENT TIMES AND USES THAT INFORMATION TO OBTAIN A VECTOR QUANTITY WHOSE MAGNITUDE TELLS ONE HOW FAST THE OBJECT IS MOVING AND WHOSE DIRECTION INDICATES THE DIRECTION THAT THE OBJECT IS MOVING.'''This page defines and describes velocity. Be sure to distinguish between velocity and [[Speed|speed]]. See [[Speed vs Velocity]] for similarities and differences.

==Main Idea==

Velocity is a [[Vectors|vector]] quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol <math>\vec{v}</math> or v, as opposed to <math>v</math>, which denotes [[Speed|speed]]. Velocity is defined as the rate of change of position with respect to time. In calculus terms, it is the time derivative of the position vector. The magnitude of a velocity vector is speed, and the direction of the vector is the direction of travel. Velocity is an instantaneous value, so it may change over time. Several properties in physics, such as [[Linear Momentum|momentum]] and [[Magnetic Force|magnetic force]], are functions of velocity. Velocity is given in unit distance per unit time. The [[SI Units|SI unit]] for velocity is the meter per second (m/s).

===A Mathematical Model===

Instantaneous velocity <math>\vec{v}</math> is defined as:

<math>\vec{v} = \frac{d\vec{r}}{dt}</math>

where <math>\vec{r}</math> is a position vector and <math>t</math> is time.

====Average Velocity====

Since velocity is an instantaneous quantity, it can change over time. Over any given time interval, there is an average velocity value denoted <math>\vec{v}_{avg}</math>. The average velocity over an interval of time <math>\Delta t</math> is given by

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

where <math>\Delta \vec{r}</math> is the displacement (change in position) over that time interval (<math>\Delta \vec{r} = \vec{r}_f - \vec{r}_i</math>).

Average velocity is often confused with average speed. A section on the [[Speed vs Velocity]] page goes into depth about the difference.

As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average velocity over that time interval approaches the value of the instantaneous velocity at any point in time within that interval.

If velocity is constant over some time interval, the average velocity over that time interval equals that constant velocity.

Here is another equation giving average velocity, this time in terms of initial velocity <math>\vec{v}_i</math> and final velocity <math>\vec{v}_f</math>:

<math>\vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2}</math>

Unlike the first equation, this equation is only true if [[Acceleration|acceleration]] is constant.

====Derivative Relationships====

Velocity is the time derivative of position:

<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.

[[Acceleration]], in turn, is the time derivative of velocity:

<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>

====Integral Relationships====

Position is the time integral of velocity:

<math>\vec{r}(t) = \int \vec{v}(t) \ dt</math>.

Integrating the velocity function as a definite integral from one time coordinate to another yields the displacement of the position over that time interval:

<math>\Delta \vec{r}_{t_1,t_2} = \int_{t_1}^{t_2} \vec{v}(t) \ dt</math>.

Velocity is, in turn, the time integral of acceleration:

<math>\vec{v}(t) = \int \vec{a}(t) \ dt</math>.

====Kinematic Equations====

The kinematic equations can be derived from the derivative and integral relationships between acceleration, velocity, and displacement. For the equations and more information, view the [[Kinematics]] page.

====In Physics====

According to [[Newton's First Law of Motion]], the velocity of an object must remain constant (whether that velocity is zero or nonzero) unless a force acts on it. [[Newton's Second Law: the Momentum Principle]] describes how velocity changes over time as a result of forces.

===A Computational Model===

In VPython simulations of physical systems, the [[Iterative Prediction]] algorithm is often used to move objects with specified velocities and evolve those velocities over time. The program below is an example of such a simulation. In this program, the ball's velocity is represented by a purple arrow.

[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]

Click "view this program" in the top left corner to view the source code.

==Examples==

===Simple===

At time t=0s, a particle positioned at (4,-1)m began to move. It moved in a straight line at a constant speed until time t=3s, at which point it had arrived at the position (-2,8)m. What was the velocity of the particle at time t=2s?

Solution:

Since the particle moved at a constant speed in a constant direction, the velocity of the particle must have been constant. That is, the velocity of the particle at time t=2 must be the same as its velocity at any other time between t=0 and t=3. This constant velocity must also be the average velocity over the time interval from t=0 to t=3, so let us use the average velocity formula:

<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>

<math>\vec{v}_{avg} = \frac{<-2, 8> - <4,-1>}{3}</math>

<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>

<math>\vec{v}_{avg} = <-2, 3></math>m/s

therefore

<math>\vec{v}(t=2) = <-2, 3></math>m/s

===Middling===

A particle's position as a function of time is as follows: <math>\vec{r}(t) = <6\sin(2 \pi t), -6t^3 + 10, e^{8t}></math>

What is the particle's velocity as a function of time?

Solution:

<math>\vec{v} = \frac{d \vec{r}}{dt}</math>

<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>

===Difficult===

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity <math>v</math> varies over time, obeying the following function:

<math>
v(t) =
\begin{cases}
4-2t & \text{if $t\leq 3$} \\
-2 & \text{if $t>3$}
\end{cases}
</math>m/s

What is its position as a function of time after time t=0?

Solution:

<math>x(t) = x(0) + \int_0^t v(t')dt'</math> (<math>t'</math> is a "dummy variable" since <math>t</math> is already our limit of integration)

for <math>t\leq 3</math>,

<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>

<math>x(t) = 2 + [4t'-t'^2]_0^t</math>

<math>x(t) = 2 + 4t - t^2</math>

for <math>t>3</math>,

<math>x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt'</math>

<math>x(t) = 5 + \int_3^t -2 dt'</math>

<math>x(t) = 5 + [-2t']_3^t</math>

<math>x(t) = 5 -2t + 6</math>

<math>x(t) = 11 - 2t</math>

final answer:

<math>
x(t) =
\begin{cases}
-t^2 + 4t + 2 & \text{if $t\leq 3$} \\
11-2t & \text{if $t>3$}
\end{cases}
</math>m

==Connectedness==

Since the goal of the field of mechanics is to predict the motion of systems, velocity is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Linear Momentum]], [[Total Angular Momentum|Angular Momentum]], and [[Magnetic Force]] depend on the velocities of objects.

The use of velocity in calculations has a wide range of industrial applications including calculating travel time, determining safe operating speeds of vehicles and other machines, vehicle collision testing, determining the inertial forces experienced by roller coaster riders, and the design of equipment such as radar guns. The velocities of moving parts are important in virtually every type of machinery.

==See Also==

*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]

===External links===

[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]

[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average Velocity]

[https://www.youtube.com/watch?v=Bxp0AWhs57g YouTube video explaining average vs instantaneous velocity]

==References==

1. Chabay, Ruth W., and Bruce A. Sherwood. Matter and Interactions. Hoboken, NJ: Wiley, 2011. Print.

2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.

[[Category:Properties of Matter]]

Bohr Model

2021-11-28T13:57:37Z

Vkumar350:

This page gives basic information about the Bohr model of the atom and the quantization of electron angular momentum. These concepts are the basis of modern quantum physics and thus are essential to master before progressing to more complex quantum theories and principles.
==Main Idea==

[[File: Bohrmodel2.png|frame|right|A visualization of the Bohr model and the hydrogen spectrum]] The Bohr model of the atom was proposed by Niels Bohr in 1913, aas an expansion on and correction of the Rutherford model. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus containing most of the atom's mass, as Rutherford had done. The Bohr Model's incorporation of quantum theory set it apart from other models; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has major shortcomings once expanded to other atoms and more complex real world situations.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun. Holding these electrons in these orbits are electrostatic forces rather than gravity.
*The energy of orbiting electrons is negative, in order to free an electron from the atom's orbit you must bring its energy to 0.
* The energy of electrons is directly related to their distance from the nucleus and which energy level they occupy at that distance. The further away the electron, the more energy it has.
* When electrons gain or lose energy they jump from one orbit to another. The energy is quantized - the orbitals have discreet radii or exact distances from the nucleus where electrons are allowed to exist, which Bohr called "stationary orbits."
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of a photon. All photons are produced by an electron transitioning to a lower energy level, or smaller radius. Conversely, an input of energy is required to transition an electron to a higher energy level. This quantized energy - in both cases - is equal to the difference between the respective energies of the orbits.
[[File:Stairsbohr.jpg]]

Stairs are a great way to visualize quantized energy. When you're going up stairs, you can only be standing on the steps, and not anywhere in between the steps. Similarly, energy can only be absorbed or emitted in specific quanta. Energy is required to go up the stairs, and energy is gained when jumping down from one stair to the next (in the electron's case, this energy is released as a photon - sometimes as visible light!).

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>

h is known as Planck's constant, which is a physical constant that is essential in quantum mechanical calculations.

==== Angular Momentum Is Quantized ====
Bohr assumed that electrons in orbit are only allowed very specific values for the magnitude of their [http://physicsbook.gatech.edu/Translational_Angular_Momentum angular momentum], specifically integer multiples of ħ given by:

<math> |\vec L_{trans,C}| = rp = Nħ</math>

L is the angular momentum of the electron,
p is the [[Linear Momentum]] of the electron,
r is the radius of the electron's orbit,
and N is an integer (1,2,3, ...).

We can derive the equation for r, the allowed Bohr radii for electron orbits for hydrogen, which has one electron and one proton.

1) The [[Electric Force]] the proton exerts on the electron is calculated using:
<math> F_{el} = {\frac{e^2}{4π ε_{0} r^2}} </math>

2) Applying concepts from the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε_{0} r^2}} </math>

3) Substituting for the relation between momentum and velocity (<math> v = {\frac{p} {m_{e}}} </math>) where <math> m_{e} </math> is the mass of an electron:
<math> {\frac{|p|}{r}} *{\frac{|p|}{m_{e}}} = {\frac{e^2}{4π ε_{0} r^2}} </math>

4) Substituting in Bohr's conditions for the magnitude of <math>p</math>:
<math> {\frac{N^2 ħ^2}{m_{e}r^3}} = {\frac{e^2}{4π ε_{0} r^2}} </math>

5) Solving for the allowed radii
<math> r = {\frac{4π ε_{0} ħ^2N^2}{m_{e}e^2}}</math>
where N = 1,2,3,...

6) This result is often simplified to
<math> r_{n} = a_{0}n^2 </math>
where <math> a_{0} = {\frac{4π ε_{0}ħ^2}{m_{e}e^2}} = 0.0529</math> nm and n = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model, and is the formula most helpful for simple calculations involving quantum mechanics.

<math> E = K + U_{electromagnetic} </math>

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{ħ}}}{N^2}}</math>

which simplifies to

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3

====Wavelengths====
In the early 1900's, light had been observed to have the properties of both a particle and a wave. In 1924, [https://en.wikipedia.org/wiki/Louis_de_Broglie Louis de Broglie], a French physicist, hypothesized all matter holds properties of waves in his thesis Recherches sur la théorie des quanta (Research on the Theory of the Quanta). According to de Broglie, there is an inverse relationship between momentum and wavelength.

The de Broglie relationship can tell us about the wavelength associated with the electron and can tell us the energy that will be released in photons, or particles of energy:
<math>λ = \frac{h}{mv}</math>
or
<math>λ = \frac{h}{p}</math>

h is Planck's constant (6.63x10e-34 joules.sec), and v is the frequency or the velocity of the disturbance in the medium of propagation.

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

Derivation of de Broglie's relationship:

E = energy, m = mass, c = speed of light,

Assuming that the two energies would be equal or in practical units:

1) <math> mc^2 = hv </math>

Since particles do not necessarily travel with the speed of light,

2) <math> mv^2 = hv = mv^2 = \frac{hv}{λ} </math>
(using <math> hv = \frac{hc}{λ}</math>)

Finally

3) <math> λ = \frac{hv}{mv^2} = \frac{h}{mv} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]

In this visualization an electron in orbit around a hydrogen nucleus progresses upwards though the available energy levels of the Bohr Model. The accompanying graph highlights the energy corresponding to each orbit with respect to the distance between the electron and the hydrogen nucleus. The greater the distance, the less negative the electron's energy and thus the less energy that must be added to free the electron from its orbit. To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here].

[[File:bohr3.gif]]

You will also find a graph of Total Energy (eV), Kinetic Energy, and Potential Energy, with each jump representing the electron transitioning to a orbit.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3. Assuming N = 1 is the lowest energy state, calculate the K+U (energy of electron) in electron volts for this atomic hydrogen energy state.

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U (a sum of the kinetic energy and the potential electromagnetic energy of the photon) = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

Below is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

[[Image:Graphproblem.png]]

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation emitted from this transition? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
1.
I chose this topic because I have unintentional conducted research on it outside of a classroom environment that was driven by a question that I have asked myself for as long as I can remember: What ''is'' light? I couldn't touch it, I couldn't produce it myself, and I could not explain where light comes from. I am so thankful that I grew up in the time where a) I could hop on the internet to see if there was an answer to my question and b) that the answer exists. We know what light is: photons of a wavelength in the visible spectrum, and we know where light comes from: energy released by electrons dropping in levels of orbit. I loved the introductory internet research I did on this subject, and it introduced me to the bizarre world of quantum physics, which I continue to be puzzled by and curious about. I also chose this topic because of my admiration for the man behind it, Niels Bohr. He was a man on the cutting edge of science who pushed our understanding of the universe by leaps and bounds by unveiling aspects of the microscopic universe.

2.
As an industrial engineering major, it is pretty difficult to quantify how a concept of physics, especially one as specific and complex as Bohr's atomic model, will directly connect to my major. This should not stop me from seeking knowledge about it to possess a more well rounded knowledge of the world. Still, it isn't hard to find applications. If I am working to optimize the production of a laser engraving facility, for example, having an understanding of the basic dynamics behind the lasers and machinery will help me to have a more level conversation with the engineers developing and maintaining the lasers, and will prevent any massive rifts of understanding between the engineering side of the facility with the men working on the business end of the operation.

3.
The industrial application of the concepts covered on this page are enormous in today's day and age. We are in the midst of an energy revolution, with solar energy looking to replace fossil fuels as the primary source of energy to the world. Understanding the energy of photons and light waves emitted from the sun is essential to the ongoing process making solar energy a more cost effective alternative to traditional fossil fuels. Once this optimization occurs, the planet will have a new primary source of energy with cleanliness and availability that is absolutely unprecedented.

==History==
[[File:PlumPuddingModel ManyCorpuscles.png|PlumPuddingModel ManyCorpuscles|left|frame|Plum pudding model, which Rutherford's model expanded on.]][[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right|frame|Niels Bohr]] To understand how revolutionary Bohr's theories and advancements were, one must have a general understanding of the accepted atomic model at the time, the Rutherford model. The Rutherford model was created in 1911 by New Zealand born chemist [https://en.wikipedia.org/wiki/Ernest_Rutherford Ernest Rutherford]. The dominant model for the structure of the atom before Rutherford's breakthrough, was the plum pudding model. While this model correctly surmised that atoms are constructed from constituents of both positive and negative charge and that the negatively charged components were quite small relative to the atom, the plum pudding model depicted electrons as stationary, lodged in place in a substance believed to constitute most of the space an atom occupies.
After conducting one of the most famous experiments in the world of physics, the [https://en.wikipedia.org/wiki/Geiger%E2%80%93Marsden_experiment Geiger-Marsden experiment], more commonly known as the gold foil experiment, Rutherford realized that the majority of the atom is empty space with the mass of the atom existing predominantly in a small volume in the center of the atom. Thus, Rutherford is credited with the discovery of the atomic nucleus. Despite its numerous breakthroughs, the Rutherford model was neither perfect nor complete. Rutherford proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]
[[File:RutherfordModel2.png|frame|right|Rutherford's model of the atom, which Bohr expanded on.]]
Niels Bohr, a physicist from Denmark, was able to explain the true nature of electrons by improving on the Rutherford model of the atom. In 1911 Bohr traveled to England in order to study the structure of atoms and molecules. There, he attended lectures on electromagnetism and worked with Rutherford and other scientists such as J. J. Thomson. When he returned to Denmark in 1912, Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. Later on, scientists such as Werner Heisenberg and Erwin Schrödinger worked to improve upon this model.[9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum, not considering more advanced quantum theories and properties that have been discovered since the Bohr first theorized the model. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom. The current working model of the atom is the quantum mechanical model of the atom, which displays the electron orbits shapes very different from the orbits of the solar system and accounts for the fact that the electrons do not exist in one specific location in the orbit, but rather exists at many certain locations in the orbital as probabilities and frequencies. See the picture below for a visual representation of this quantum mechanical atomic model.
[[File:Quantumorbitals.jpeg]]

== See also ==

===Further Reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

Why Bohr's model explains everything around us:

http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/

==References==

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
[https://science.howstuffworks.com/atom9.htm]
Note: All images on this page are either free for commercial use (with no attribution required) or made by myself.
[[Category:Energy]]

Bohr Model

2021-11-28T13:53:51Z

Vkumar350:

This page gives basic information about the Bohr model of the atom and the quantization of electron angular momentum. These concepts are the basis of modern quantum physics and thus are essential to master before progressing to more complex quantum theories and principles.
==Main Idea==

CLAIMED BY VIBHAV KUMAR (FALL 2021)

[[File: Bohrmodel2.png|frame|right|A visualization of the Bohr model and the hydrogen spectrum]] The Bohr model of the atom was proposed by Niels Bohr in 1913, aas an expansion on and correction of the Rutherford model. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus containing most of the atom's mass, as Rutherford had done. The Bohr Model's incorporation of quantum theory set it apart from other models; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has major shortcomings once expanded to other atoms and more complex real world situations.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun. Holding these electrons in these orbits are electrostatic forces rather than gravity.
*The energy of orbiting electrons is negative, in order to free an electron from the atom's orbit you must bring its energy to 0.
* The energy of electrons is directly related to their distance from the nucleus and which energy level they occupy at that distance. The further away the electron, the more energy it has.
* When electrons gain or lose energy they jump from one orbit to another. The energy is quantized - the orbitals have discreet radii or exact distances from the nucleus where electrons are allowed to exist, which Bohr called "stationary orbits."
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of a photon. All photons are produced by an electron transitioning to a lower energy level, or smaller radius. Conversely, an input of energy is required to transition an electron to a higher energy level. This quantized energy - in both cases - is equal to the difference between the respective energies of the orbits.
[[File:Stairsbohr.jpg]]

Stairs are a great way to visualize quantized energy. When you're going up stairs, you can only be standing on the steps, and not anywhere in between the steps. Similarly, energy can only be absorbed or emitted in specific quanta. Energy is required to go up the stairs, and energy is gained when jumping down from one stair to the next (in the electron's case, this energy is released as a photon - sometimes as visible light!).

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>

h is known as Planck's constant, which is a physical constant that is essential in quantum mechanical calculations.

==== Angular Momentum Is Quantized ====
Bohr assumed that electrons in orbit are only allowed very specific values for the magnitude of their [http://physicsbook.gatech.edu/Translational_Angular_Momentum angular momentum], specifically integer multiples of ħ given by:

<math> |\vec L_{trans,C}| = rp = Nħ</math>

L is the angular momentum of the electron,
p is the [[Linear Momentum]] of the electron,
r is the radius of the electron's orbit,
and N is an integer (1,2,3, ...).

We can derive the equation for r, the allowed Bohr radii for electron orbits for hydrogen, which has one electron and one proton.

1) The [[Electric Force]] the proton exerts on the electron is calculated using:
<math> F_{el} = {\frac{e^2}{4π ε_{0} r^2}} </math>

2) Applying concepts from the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε_{0} r^2}} </math>

3) Substituting for the relation between momentum and velocity (<math> v = {\frac{p} {m_{e}}} </math>) where <math> m_{e} </math> is the mass of an electron:
<math> {\frac{|p|}{r}} *{\frac{|p|}{m_{e}}} = {\frac{e^2}{4π ε_{0} r^2}} </math>

4) Substituting in Bohr's conditions for the magnitude of <math>p</math>:
<math> {\frac{N^2 ħ^2}{m_{e}r^3}} = {\frac{e^2}{4π ε_{0} r^2}} </math>

5) Solving for the allowed radii
<math> r = {\frac{4π ε_{0} ħ^2N^2}{m_{e}e^2}}</math>
where N = 1,2,3,...

6) This result is often simplified to
<math> r_{n} = a_{0}n^2 </math>
where <math> a_{0} = {\frac{4π ε_{0}ħ^2}{m_{e}e^2}} = 0.0529</math> nm and n = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model, and is the formula most helpful for simple calculations involving quantum mechanics.

<math> E = K + U_{electromagnetic} </math>

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{ħ}}}{N^2}}</math>

which simplifies to

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3

====Wavelengths====
In the early 1900's, light had been observed to have the properties of both a particle and a wave. In 1924, [https://en.wikipedia.org/wiki/Louis_de_Broglie Louis de Broglie], a French physicist, hypothesized all matter holds properties of waves in his thesis Recherches sur la théorie des quanta (Research on the Theory of the Quanta). According to de Broglie, there is an inverse relationship between momentum and wavelength.

The de Broglie relationship can tell us about the wavelength associated with the electron and can tell us the energy that will be released in photons, or particles of energy:
<math>λ = \frac{h}{mv}</math>
or
<math>λ = \frac{h}{p}</math>

h is Planck's constant (6.63x10e-34 joules.sec), and v is the frequency or the velocity of the disturbance in the medium of propagation.

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

Derivation of de Broglie's relationship:

E = energy, m = mass, c = speed of light,

Assuming that the two energies would be equal or in practical units:

1) <math> mc^2 = hv </math>

Since particles do not necessarily travel with the speed of light,

2) <math> mv^2 = hv = mv^2 = \frac{hv}{λ} </math>
(using <math> hv = \frac{hc}{λ}</math>)

Finally

3) <math> λ = \frac{hv}{mv^2} = \frac{h}{mv} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]

In this visualization an electron in orbit around a hydrogen nucleus progresses upwards though the available energy levels of the Bohr Model. The accompanying graph highlights the energy corresponding to each orbit with respect to the distance between the electron and the hydrogen nucleus. The greater the distance, the less negative the electron's energy and thus the less energy that must be added to free the electron from its orbit. To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here].

[[File:bohr3.gif]]

You will also find a graph of Total Energy (eV), Kinetic Energy, and Potential Energy, with each jump representing the electron transitioning to a orbit.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3. Assuming N = 1 is the lowest energy state, calculate the K+U (energy of electron) in electron volts for this atomic hydrogen energy state.

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U (a sum of the kinetic energy and the potential electromagnetic energy of the photon) = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

Below is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

[[Image:Graphproblem.png]]

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation emitted from this transition? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
1.
I chose this topic because I have unintentional conducted research on it outside of a classroom environment that was driven by a question that I have asked myself for as long as I can remember: What ''is'' light? I couldn't touch it, I couldn't produce it myself, and I could not explain where light comes from. I am so thankful that I grew up in the time where a) I could hop on the internet to see if there was an answer to my question and b) that the answer exists. We know what light is: photons of a wavelength in the visible spectrum, and we know where light comes from: energy released by electrons dropping in levels of orbit. I loved the introductory internet research I did on this subject, and it introduced me to the bizarre world of quantum physics, which I continue to be puzzled by and curious about. I also chose this topic because of my admiration for the man behind it, Niels Bohr. He was a man on the cutting edge of science who pushed our understanding of the universe by leaps and bounds by unveiling aspects of the microscopic universe.

2.
As an industrial engineering major, it is pretty difficult to quantify how a concept of physics, especially one as specific and complex as Bohr's atomic model, will directly connect to my major. This should not stop me from seeking knowledge about it to possess a more well rounded knowledge of the world. Still, it isn't hard to find applications. If I am working to optimize the production of a laser engraving facility, for example, having an understanding of the basic dynamics behind the lasers and machinery will help me to have a more level conversation with the engineers developing and maintaining the lasers, and will prevent any massive rifts of understanding between the engineering side of the facility with the men working on the business end of the operation.

3.
The industrial application of the concepts covered on this page are enormous in today's day and age. We are in the midst of an energy revolution, with solar energy looking to replace fossil fuels as the primary source of energy to the world. Understanding the energy of photons and light waves emitted from the sun is essential to the ongoing process making solar energy a more cost effective alternative to traditional fossil fuels. Once this optimization occurs, the planet will have a new primary source of energy with cleanliness and availability that is absolutely unprecedented.

==History==
[[File:PlumPuddingModel ManyCorpuscles.png|PlumPuddingModel ManyCorpuscles|left|frame|Plum pudding model, which Rutherford's model expanded on.]][[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right|frame|Niels Bohr]] To understand how revolutionary Bohr's theories and advancements were, one must have a general understanding of the accepted atomic model at the time, the Rutherford model. The Rutherford model was created in 1911 by New Zealand born chemist [https://en.wikipedia.org/wiki/Ernest_Rutherford Ernest Rutherford]. The dominant model for the structure of the atom before Rutherford's breakthrough, was the plum pudding model. While this model correctly surmised that atoms are constructed from constituents of both positive and negative charge and that the negatively charged components were quite small relative to the atom, the plum pudding model depicted electrons as stationary, lodged in place in a substance believed to constitute most of the space an atom occupies.
After conducting one of the most famous experiments in the world of physics, the [https://en.wikipedia.org/wiki/Geiger%E2%80%93Marsden_experiment Geiger-Marsden experiment], more commonly known as the gold foil experiment, Rutherford realized that the majority of the atom is empty space with the mass of the atom existing predominantly in a small volume in the center of the atom. Thus, Rutherford is credited with the discovery of the atomic nucleus. Despite its numerous breakthroughs, the Rutherford model was neither perfect nor complete. Rutherford proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]
[[File:RutherfordModel2.png|frame|right|Rutherford's model of the atom, which Bohr expanded on.]]
Niels Bohr, a physicist from Denmark, was able to explain the true nature of electrons by improving on the Rutherford model of the atom. In 1911 Bohr traveled to England in order to study the structure of atoms and molecules. There, he attended lectures on electromagnetism and worked with Rutherford and other scientists such as J. J. Thomson. When he returned to Denmark in 1912, Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. Later on, scientists such as Werner Heisenberg and Erwin Schrödinger worked to improve upon this model.[9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum, not considering more advanced quantum theories and properties that have been discovered since the Bohr first theorized the model. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom. The current working model of the atom is the quantum mechanical model of the atom, which displays the electron orbits shapes very different from the orbits of the solar system and accounts for the fact that the electrons do not exist in one specific location in the orbit, but rather exists at many certain locations in the orbital as probabilities and frequencies. See the picture below for a visual representation of this quantum mechanical atomic model.
[[File:Quantumorbitals.jpeg]]

== See also ==

===Further Reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

Why Bohr's model explains everything around us:

http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/

==References==

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
[https://science.howstuffworks.com/atom9.htm]
Note: All images on this page are either free for commercial use (with no attribution required) or made by myself.
[[Category:Energy]]

Bohr Model

2021-11-28T13:53:36Z

Vkumar350:

This page gives basic information about the Bohr model of the atom and the quantization of electron angular momentum. These concepts are the basis of modern quantum physics and thus are essential to master before progressing to more complex quantum theories and principles.
==Main Idea==

CLAIMED BY VIBHAV KUMAR

[[File: Bohrmodel2.png|frame|right|A visualization of the Bohr model and the hydrogen spectrum]] The Bohr model of the atom was proposed by Niels Bohr in 1913, aas an expansion on and correction of the Rutherford model. His model depicted atoms as having negatively charged electrons which orbited as small, positively charged nucleus containing most of the atom's mass, as Rutherford had done. The Bohr Model's incorporation of quantum theory set it apart from other models; in this model, electrons can only exist in discrete energy levels, which are quantized. This model is very simplistic and is useful in introducing students to quantum mechanics. Although this model successfully predicts energy levels of the hydrogen atom, it has major shortcomings once expanded to other atoms and more complex real world situations.
===Bohr's Assumptions===
* Electrons travel in a circular orbit around the nucleus, similar to how planets orbit around the sun. Holding these electrons in these orbits are electrostatic forces rather than gravity.
*The energy of orbiting electrons is negative, in order to free an electron from the atom's orbit you must bring its energy to 0.
* The energy of electrons is directly related to their distance from the nucleus and which energy level they occupy at that distance. The further away the electron, the more energy it has.
* When electrons gain or lose energy they jump from one orbit to another. The energy is quantized - the orbitals have discreet radii or exact distances from the nucleus where electrons are allowed to exist, which Bohr called "stationary orbits."
* When an excited electron returns back to its ground state, then it releases the energy that it absorbed, in the form of a photon. All photons are produced by an electron transitioning to a lower energy level, or smaller radius. Conversely, an input of energy is required to transition an electron to a higher energy level. This quantized energy - in both cases - is equal to the difference between the respective energies of the orbits.
[[File:Stairsbohr.jpg]]

Stairs are a great way to visualize quantized energy. When you're going up stairs, you can only be standing on the steps, and not anywhere in between the steps. Similarly, energy can only be absorbed or emitted in specific quanta. Energy is required to go up the stairs, and energy is gained when jumping down from one stair to the next (in the electron's case, this energy is released as a photon - sometimes as visible light!).

==Application==
===A Mathematical Model===
==== The Angular Momentum Quantum ====
<math>ħ = h/2π =1.05*10^{-34} J*s </math>

h is known as Planck's constant, which is a physical constant that is essential in quantum mechanical calculations.

==== Angular Momentum Is Quantized ====
Bohr assumed that electrons in orbit are only allowed very specific values for the magnitude of their [http://physicsbook.gatech.edu/Translational_Angular_Momentum angular momentum], specifically integer multiples of ħ given by:

<math> |\vec L_{trans,C}| = rp = Nħ</math>

L is the angular momentum of the electron,
p is the [[Linear Momentum]] of the electron,
r is the radius of the electron's orbit,
and N is an integer (1,2,3, ...).

We can derive the equation for r, the allowed Bohr radii for electron orbits for hydrogen, which has one electron and one proton.

1) The [[Electric Force]] the proton exerts on the electron is calculated using:
<math> F_{el} = {\frac{e^2}{4π ε_{0} r^2}} </math>

2) Applying concepts from the momentum principle and curving motion:
<math>|F_{perpendicular}| = {\frac{|p| |v|}{r}} = {\frac{e^2}{4π ε_{0} r^2}} </math>

3) Substituting for the relation between momentum and velocity (<math> v = {\frac{p} {m_{e}}} </math>) where <math> m_{e} </math> is the mass of an electron:
<math> {\frac{|p|}{r}} *{\frac{|p|}{m_{e}}} = {\frac{e^2}{4π ε_{0} r^2}} </math>

4) Substituting in Bohr's conditions for the magnitude of <math>p</math>:
<math> {\frac{N^2 ħ^2}{m_{e}r^3}} = {\frac{e^2}{4π ε_{0} r^2}} </math>

5) Solving for the allowed radii
<math> r = {\frac{4π ε_{0} ħ^2N^2}{m_{e}e^2}}</math>
where N = 1,2,3,...

6) This result is often simplified to
<math> r_{n} = a_{0}n^2 </math>
where <math> a_{0} = {\frac{4π ε_{0}ħ^2}{m_{e}e^2}} = 0.0529</math> nm and n = 1,2,3,...

Additionally, the formula for energy of hydrogen atom of different levels is also derived from this model, and is the formula most helpful for simple calculations involving quantum mechanics.

<math> E = K + U_{electromagnetic} </math>

1)
<math> E = {\frac{mv^2}{2}} - {\frac{{\frac{1}{2}}*{\frac{1}{4π ε0}}*{\frac{me^2}{ħ}}}{N^2}}</math>

which simplifies to

2)
<math> E = {\frac{13.6 eV}{N^2}}</math> where N = 1,2,3

====Wavelengths====
In the early 1900's, light had been observed to have the properties of both a particle and a wave. In 1924, [https://en.wikipedia.org/wiki/Louis_de_Broglie Louis de Broglie], a French physicist, hypothesized all matter holds properties of waves in his thesis Recherches sur la théorie des quanta (Research on the Theory of the Quanta). According to de Broglie, there is an inverse relationship between momentum and wavelength.

The de Broglie relationship can tell us about the wavelength associated with the electron and can tell us the energy that will be released in photons, or particles of energy:
<math>λ = \frac{h}{mv}</math>
or
<math>λ = \frac{h}{p}</math>

h is Planck's constant (6.63x10e-34 joules.sec), and v is the frequency or the velocity of the disturbance in the medium of propagation.

This equation is used to help derive the equation for the angular momentum of an electron in orbit
<math> L = \frac{nh}{2π} </math>

Derivation of de Broglie's relationship:

E = energy, m = mass, c = speed of light,

Assuming that the two energies would be equal or in practical units:

1) <math> mc^2 = hv </math>

Since particles do not necessarily travel with the speed of light,

2) <math> mv^2 = hv = mv^2 = \frac{hv}{λ} </math>
(using <math> hv = \frac{hc}{λ}</math>)

Finally

3) <math> λ = \frac{hv}{mv^2} = \frac{h}{mv} </math>

===A Computational Model===

[[File:bohr2.gif|upright]]

In this visualization an electron in orbit around a hydrogen nucleus progresses upwards though the available energy levels of the Bohr Model. The accompanying graph highlights the energy corresponding to each orbit with respect to the distance between the electron and the hydrogen nucleus. The greater the distance, the less negative the electron's energy and thus the less energy that must be added to free the electron from its orbit. To interact with this glowscript visualization of the Bohr Model, click [http://www.glowscript.org/#/user/matterandinteractions/folder/matterandinteractions/program/08-Bohr-levels here].

[[File:bohr3.gif]]

You will also find a graph of Total Energy (eV), Kinetic Energy, and Potential Energy, with each jump representing the electron transitioning to a orbit.

==Examples==

===Simple===
Find the magnitude of the translational angular momentum of an electron when a hydrogen atom is in its 2nd excited state above the ground state.

--

We know that the only possible states of the hydrogen atom are those when the electron's translational angular momentum is an integer multiple of ħ.
<math>|\vec L_{trans,c}| = Nħ </math>
For the 2nd excited state, N = 3
Now just plug the numbers in
<math>|\vec L_{trans,c}| = (3)(1.05*10^{-34} J*s) = 3.15*10^{-34} J*s</math>

===Middling===
[6]A hydrogen atom is in state N = 3. Assuming N = 1 is the lowest energy state, calculate the K+U (energy of electron) in electron volts for this atomic hydrogen energy state.

1)
<math>E(3) = {\frac{-13.6 eV}{3^2}}</math> = -1.51 Joules

2)
<math>E(1) = {\frac{-13.6 eV}{1^2}}</math> = -13.6 Joules

3)
K+U (a sum of the kinetic energy and the potential electromagnetic energy of the photon) = energy of photon = <math>E(1) - E(3) = {\frac{-13.6 eV}{3^2}} - {\frac{-13.6 eV}{1^2}}</math> = 12.09 Joules

Below is the graph of E as the hydrogen atom goes from N = 3 to N = 1.

[[Image:Graphproblem.png]]

===Difficult===
Hydrogen has been detected transitioning from the 101st to the 100th energy levels. What is the wavelength of the radiation emitted from this transition? Where in the electromagnetic spectrum is this emission?

To solve this problem, we first need to use formulas derived from Bohr Model of hydrogen atom. It is <math>E = {\frac{-13.6 eV}{N^2}}</math>

Then solve for the wavelength using formula from Electromagnetic Wave Theory.

[[Image:Screen Shot 2015-12-01 at 8.21.56 PM.png|200x250px|]]

This wavelength falls in the microwave portion of the electromagnetic spectrum.

==Connectedness==
1.
I chose this topic because I have unintentional conducted research on it outside of a classroom environment that was driven by a question that I have asked myself for as long as I can remember: What ''is'' light? I couldn't touch it, I couldn't produce it myself, and I could not explain where light comes from. I am so thankful that I grew up in the time where a) I could hop on the internet to see if there was an answer to my question and b) that the answer exists. We know what light is: photons of a wavelength in the visible spectrum, and we know where light comes from: energy released by electrons dropping in levels of orbit. I loved the introductory internet research I did on this subject, and it introduced me to the bizarre world of quantum physics, which I continue to be puzzled by and curious about. I also chose this topic because of my admiration for the man behind it, Niels Bohr. He was a man on the cutting edge of science who pushed our understanding of the universe by leaps and bounds by unveiling aspects of the microscopic universe.

2.
As an industrial engineering major, it is pretty difficult to quantify how a concept of physics, especially one as specific and complex as Bohr's atomic model, will directly connect to my major. This should not stop me from seeking knowledge about it to possess a more well rounded knowledge of the world. Still, it isn't hard to find applications. If I am working to optimize the production of a laser engraving facility, for example, having an understanding of the basic dynamics behind the lasers and machinery will help me to have a more level conversation with the engineers developing and maintaining the lasers, and will prevent any massive rifts of understanding between the engineering side of the facility with the men working on the business end of the operation.

3.
The industrial application of the concepts covered on this page are enormous in today's day and age. We are in the midst of an energy revolution, with solar energy looking to replace fossil fuels as the primary source of energy to the world. Understanding the energy of photons and light waves emitted from the sun is essential to the ongoing process making solar energy a more cost effective alternative to traditional fossil fuels. Once this optimization occurs, the planet will have a new primary source of energy with cleanliness and availability that is absolutely unprecedented.

==History==
[[File:PlumPuddingModel ManyCorpuscles.png|PlumPuddingModel ManyCorpuscles|left|frame|Plum pudding model, which Rutherford's model expanded on.]][[Image:200px-Niels_Bohr_Date_Unverified_LOC.jpg |right|frame|Niels Bohr]] To understand how revolutionary Bohr's theories and advancements were, one must have a general understanding of the accepted atomic model at the time, the Rutherford model. The Rutherford model was created in 1911 by New Zealand born chemist [https://en.wikipedia.org/wiki/Ernest_Rutherford Ernest Rutherford]. The dominant model for the structure of the atom before Rutherford's breakthrough, was the plum pudding model. While this model correctly surmised that atoms are constructed from constituents of both positive and negative charge and that the negatively charged components were quite small relative to the atom, the plum pudding model depicted electrons as stationary, lodged in place in a substance believed to constitute most of the space an atom occupies.
After conducting one of the most famous experiments in the world of physics, the [https://en.wikipedia.org/wiki/Geiger%E2%80%93Marsden_experiment Geiger-Marsden experiment], more commonly known as the gold foil experiment, Rutherford realized that the majority of the atom is empty space with the mass of the atom existing predominantly in a small volume in the center of the atom. Thus, Rutherford is credited with the discovery of the atomic nucleus. Despite its numerous breakthroughs, the Rutherford model was neither perfect nor complete. Rutherford proposed that the emission spectrum of hydrogen would look more like a smear rather than being made up of distinct lines. However, this was flawed, as it suggested that atoms can emit energy that isn't quantized. [8]
[[File:RutherfordModel2.png|frame|right|Rutherford's model of the atom, which Bohr expanded on.]]
Niels Bohr, a physicist from Denmark, was able to explain the true nature of electrons by improving on the Rutherford model of the atom. In 1911 Bohr traveled to England in order to study the structure of atoms and molecules. There, he attended lectures on electromagnetism and worked with Rutherford and other scientists such as J. J. Thomson. When he returned to Denmark in 1912, Bohr noticed that in the atomic emissions spectrum of hydrogen, only certain colors could be seen. Thus, he theorized that electrons need to be in energy levels that are quantized. He related the energies of the colors he saw to the differences of hydrogen's energy levels. Although this model is not entirely correct, as it only applies to systems where two charged particles orbit each other, it still has many features that are very applicable to physics today. Later on, scientists such as Werner Heisenberg and Erwin Schrödinger worked to improve upon this model.[9]

== Shortcomings of the Bohr Model ==
While the Bohr model is an important predecessor to the current quantum mechanical models of the atom, it doesn't correctly describe some aspects of electron orbitals. It does not provide any reasoning as to why certain spectral lines are brighter than others. Its main shortcoming is that it violates the uncertainty principle, as it considers electrons to have a definite radius and momentum, not considering more advanced quantum theories and properties that have been discovered since the Bohr first theorized the model. This model is very basic, and in order to know more specific details about spectra and charge distribution, more calculations must be done. Many of the failures of the Bohr model can be corrected by the [http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c2 Schrodinger equation]for the hydrogen atom. The current working model of the atom is the quantum mechanical model of the atom, which displays the electron orbits shapes very different from the orbits of the solar system and accounts for the fact that the electrons do not exist in one specific location in the orbit, but rather exists at many certain locations in the orbital as probabilities and frequencies. See the picture below for a visual representation of this quantum mechanical atomic model.
[[File:Quantumorbitals.jpeg]]

== See also ==

===Further Reading===

Matter and Interactions I Modern Mechanics 4th Edition Chapter 11.10

===External links===

https://en.wikipedia.org/wiki/Bohr_model

https://en.wikipedia.org/wiki/Quantization_(physics)

Videos:

https://www.khanacademy.org/science/chemistry/electronic-structure-of-atoms/bohr-model-hydrogen/v/bohr-model-energy-levels

https://www.youtube.com/watch?v=nVW1zDPPZGM

Simulation of Bohr Model:

https://phet.colorado.edu/en/simulation/legacy/hydrogen-atom

Why Bohr's model explains everything around us:

http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/

==References==

[http://pachamamatrust.org/f2/1_K/SP_physics/H4_bohr_model_KSP.htm]
[http://chemistry.about.com/od/atomicstructure/a/bohr-model.htm]
[http://scitech.au.dk/en/roemer/apr13/bohrs-model-of-the-atom-explains-science-in-everyday-life/]
[http://www.encyclopedia.com/topic/Bohr_model.aspx]
[http://ib-physics-ii-6b-e.aspen.high.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4397333&fid=18592200]
[https://files.t-square.gatech.edu/access/content/group/gtc-80c7-d8ef-5c05-873d-52adf5b9ce5c/Lecture%20Notes/Fenton/Phys2211_C_2015_fall_week_16_xMonday.pdf]
[http://hyperphysics.phy-astr.gsu.edu/hbase/bohr.html]
[http://web.ihep.su/dbserv/compas/src/bohr13/eng.pdf]
[http://chemed.chem.purdue.edu/genchem/history/bohr.html]
[https://science.howstuffworks.com/atom9.htm]
Note: All images on this page are either free for commercial use (with no attribution required) or made by myself.
[[Category:Energy]]