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This page defines and describes velocity.
'''CLAIMED BY ANJIYA KARIM (FALL 2022)'''


==Main Idea==
==Main Idea==
Velocity, denoted by the symbol <math>\vec{v}</math>, is a vector quantity 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 momentum and kinetic energy, are functions of velocity. The most commonly used metric unit for velocity is the meter per second (m/s).
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 <b>v</b>, 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===
===A Mathematical Model===


[[File:Acceleration as derivative of velocity along trajectory.svg|right|thumb|This graphic was found [https://commons.wikimedia.org/wiki/File:Acceleration_as_derivative_of_velocity_along_trajectory.svg#/media/File:Acceleration_as_derivative_of_velocity_along_trajectory.svg here] on the Wikimedia commons, where it was uploaded by user 'Fred the Oyster.' It is provided on the commons under an [https://creativecommons.org/licenses/by-sa/4.0/ Attribution-ShareAlike 4.0 International] license. It is shown here under the same license.]]
Instantaneous velocity <math>\vec{v}</math> is defined as:


Instantaneous acceleration is defined as:
<math>\vec{v} = \frac{d\vec{r}}{dt}</math>


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


====Average Acceleration====
====Average Velocity====


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


====Derivative Relationships====
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)''.
 
 
 
 
 
 
 
 
 
 
 
 
 
====Angular Velocity====
Angular velocity, also known as rotational velocity, is how fast a body rotates around a specific center of rotation. It is denoted as omega ω. There are two ways an object can rotate:
        1) Rotating around its own center of mass. For example, spinning a ball on your finger.
        2) Rotating around a point in space. For example, the rotation of Earth around the Sun.
The higher the angular velocity, the faster the object would rotate. In order to determine the its direction, we can use the right hand rule. By curling the fingers of the right hand in the direction in which the object is turning, the thumb would end up pointing in the direction of the axis of rotation.
   
Angular Velocity Formulas
  There are two formulas for calculating angular velocity:
        1) ω=(α2−α1)/t, where α represents angles of a circle in radians and t represents time in seconds.
        2) ω=s/rxt, where s is the arc length, and r is the radius of the circle.
        3) ω=v/r, where v is the linear velocity, and r is the radius.
 
Calculating Angular Velocity:
    We can calculate the average angular velocity of the moon in its orbit around the Earth by the following method:
        -The moon takes about 27.3 days (2,358,720 seconds)to orbit around the Earth
        ω=2π/t = 2.7*10^-6/s.
 
====Velocity and Acceleration Relationship====


Acceleration is the time derivative of velocity, which is in turn the time derivative of position, so acceleration is the second time derivative of position.
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.


Recall that


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


and
Velocity is the time derivative of position:


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


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


<math>\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}</math>.
<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>


====Integral Relationships====
====Integral Relationships====


The integral of acceleration yields velocity, and the integral of velocity yields position, so the double integral of acceleration yields position.
Position is the time integral of velocity:


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


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


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


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


====Kinematic Equations====
====Kinematic Equations====
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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.
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.


====Newton's Second Law of Motion====
====In Physics====


In physics, the acceleration of particles is caused by forces. [[Newton's Second Law: the Momentum Principle]] states that
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.


<math>\vec{F}_{net} = \frac{d\vec{p}}{dt}</math>.
===A Computational Model===


From this relation, it can be derived that
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.


<math>\vec{F}_{net} = m\vec{a}</math>
[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]


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


<math>\vec{a} = \frac{\vec{F}_{net}}{m}</math>.
Here [https://www.glowscript.org/#/user/anjiyakarim/folder/MyPrograms/program/wikivelocitytimegraph] is another glow script model that represents the change in velocity of the object with respect to time. It also represents how the velocity might affect the change in position of the object.


This makes sense because acceleration should be directly proportional to force, which causes it, and inversely proportional to mass, which resists it (see [[Inertia]]). This equation is often used to find the acceleration of a particle. If that acceleration is constant, kinematic equations are then often used to determine how its position and velocity change over time.
===Velocity Time Graphs===
Velocity-Time graphs describe the motion of the objects. Here, the time(s) is often represented at the x-axis since it is the independent variable, while the velocity is represented at the y-axis. The area under these graphs represents the displacement of the object while the slope represents the acceleration. Below is an example of a velocity time graph:


==A Computational Model==
[[File:Example-velocity-time-diagramm.svg|thumb|left|Velocity-Time Graph ]]
This model defines an object and models its displacement arbitrarily with its mass in relation to time,


  m=9
  g=9.81
  t=0
  deltat=1
  positionInitial=vector(0,0,0)
  while t<6:
    positionFinal=vector(0,m*g*t,0)
    displacement=positionFinal-positionInitial
    velocity=displacement/deltat
    t=t+deltat
  print (velocity ,"is velocity")


after t=6 is reached, the updated velocity is given.
====Zero Velocity====
When a vehicle has zero velocity, it is said to be stationary. Hence, it does not cover any distance per unit time. A graph with a solid line along the x-axis at y=0 is formed.


This specific vPython example illustrates how a "while loop" may be used to update velocity, as velocity can be ever-changing as forces are applied.  
====Constant Velocity====
When a vehicle travels at a constant velocity in a particular direction, the acceleration is equal to zero. A graph for such motion consists of a flat line parallel of the x-axis. However, the line is not equal to y=0. In such situations, the vehicle is moving, but the velocity is constant at each time interval.


==Example==


A car takes 3 hours to make a 230-mile trip from Point A to Point B.  
====Increasing/Decreasing Velocity at Constant Rate====
When a vehicle starts to travel faster/slower, but the change of velocity is constant per unit time, a graph with a constant positive/negative slope is obtained: Such graph also reflects constant acceleration/deceleration of the object.


{| border="1"
====Increasing Velocity at a Non-Constant Rate ====
|+
When a vehicle starts to travel faster, and the change of velocity also increasing per unit time, a curved shaped graph pointing upwards is formed. Such graphs reflect increasing acceleration.
!  !! Hour 1 !! Hour 2 !! Hour 3
|-
! Velocity
| 80 mph north || 90 mph north || 60 mph north
|-
|}


There are two kinds of velocity in which one must consider: instantaneous velocity and average velocity.
====Decreasing Velocity at a Non-Constant Rate ====
[https://www.youtube.com/watch?v=Bxp0AWhs57g] does a good job explaining the difference between the two types of velocity
When a vehicle starts to slow down and the change in velocity is also decreasing per unit time, the graph with a curve pointing downwards in obtained. This reflects that the object is decelerating.


===Instantaneous Velocity===
==Examples==


Instantaneous velocity is the speed and direction of an object at a particular instant. Mathematically, it is the derivative of the position function at a specific point in time.
===Simple===


Given the example: Each hour, and each time point in every hour has a different instantaneous velocity.
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?


===Average Velocity===
Solution:


Average velocity is the net displacement of an object, divided by the total travel time. It is the average of all instantaneous velocities. It is important to note that as <math>{\Delta\mathit{t}}</math> gets very small, the average velocity approaches the instantaneous velocity.
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:


Given the example: The average velocity would be (230 miles/3 hours) = 76.67 mph north.
<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>


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


Acceleration is the rate of change of velocity, divided by the change in time, modeled with with the following equation:
<math>\vec{v}_{avg} = \frac{<-6, 9>}{3}</math>
:<math>\boldsymbol{a} = \frac{\Delta\boldsymbol{v}}{\Delta\mathit{t}}</math> ,  
where <math>{\Delta\boldsymbol{v}}</math> is the change of velocity of the object and <math>{\Delta\mathit{t}}</math> is the change of time.


The SI units for acceleration are ''meters per second squared (m/s/s)''. It is also a vector quantity.
<math>\vec{v}_{avg} = <-2, 3></math>m/s


Given the example: The acceleration from the 1st hour to the 2nd hour is 10 mph. This indicates a positive acceleration. The acceleration from the 2nd hour to the 3rd hour is -30 mph. This indicates a negative acceleration.
therefore


Colloquially acceleration is referred to as "speeding up" whilst "slowing down" is decelerating. Bear in mind that the direction does not have to change for deceleration to take place, it simply has to slow down.
<math>\vec{v}(t=2) = <-2, 3></math>m/s


<h2>Another Example</h2>
===Middling===
[[File:Velocity-time_graph_example.png]]


Based on what you know about velocity in relation to acceleration. During the time interval of 3-5 seconds is the object accelerating or decelerating? How about from 12-14 seconds? How do you know both of these answers?
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>


Given the Example: From 3-5 seconds, knowing that acceleration is the derivative of velocity, it can be seen that the object is accelerating, as the graph has a positive slopes.From 12-14 seconds, the graph has an increasingly negative slope, signifying deceleration towards zero.
What is the particle's velocity as a function of time?


==Momentum==
Solution:
Another application of velocity is within the realm of momentum and the Momentum Principle. momentum is defined as the mass of an object multiple by its vector velocity quantity. Like velocity momentum is a vector quantity. This quantity can be used in conjunction with change in time to see the amount of force applied on an object, and by extension its final location and velocity. This can be modeled iteratively through computer programs or be done in one calculation.


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


<h2> An Introductory Example </h2>
<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>
If a ball travels from location <2,4,6>m to <3,5,8>m in two seconds, what is its velocity?


Solution: :<math>\boldsymbol{\bar{v}} = \frac{\Delta\boldsymbol{r}}{\Delta\mathit{t}}</math>
===Difficult===
Delta r: <3,5,8>m-<2,4,6>m and delta t is equal to 2s, so velocity is equal to the vector <1,1,2>m/2s, which is equal to <0.5,0.5,1> m/s


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:


<h2> A More Difficult Example </h2>
<math>
A car is moving with a velocity of <26,87,12> m/s. If the initial location of the car is at <0,0,0>m and the final location of the car is at <39,130.5, 18> m, how many seconds did the car travel?
  v(t) =
  \begin{cases}
                                4-2t & \text{if $t\leq 3$} \\
                                  -2 & \text{if $t>3$}
  \end{cases}
</math>m/s


Assuming a constant velocity, delta r is equal to the final location of the car, since the car began at the origin. By dividing the delta r by the given velocity, a total time of two seconds of travel can be found.
What is its position as a function of time after time t=0?


Solution:


<h2>A Final Example with Application of Momentum </h2>
<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)
If a van has a mass of 1200 kilograms, and it is traveling with a velocity of magnitude 38 m/s, what is its momentum?


Its momentum is 45,600 kg*m/s. This can be obtained by multiplying the mass by the magnitude of the velocity.
for <math>t\leq 3</math>,


==Connectedness==
<math>x(t) = 2 + \int_0^t 4-2t' dt'</math>
Velocity is a very simple yet interesting concept in the way that it can be applied to many different parts of physics from something as simple as displacement. Velocity's sheer versatility as a concept and the number of things that can be derived from it, which include acceleration, momentum, and by extension, force and mass. Also because it can be related to force, it can be used, in conjunction with other types of forces to determine many things about systems.
 
<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>


Velocity is also of critical importance in calculating kinetic energy (0.5*m*v^2). Through understanding both the relationship between kinetic energy and velocity, as well as the resulting relationship between kinetic energy and total energy, concepts such as potential spring, gravitational, and electric energies may also be related back to velocity. As Physics 1 deals primarily with motion and predicting future motion, velocity is an absolutely critical tool, as it can often be used to analyze changes of external forces and predict future movement.
<math>x(t) = 5 + \int_3^t -2 dt'</math>


Velocity relates to my career aspirations in a rather interesting way. Because I plan on trying to become a trauma doctor, its easy to see the difference between high and low velocity impacts of objects of the same mass. If a low mass object is accelerating at a high enough velocity, the ramifications of its impact with the body could be vastly different than an object with a low velocity.
<math>x(t) = 5 + [-2t']_3^t</math>


An industrial application of velocity could be seen in cars and the limits of their engines. The limit to which a car engine can perform can be tested in various ways, one of them being velocity. This could be one reason why you don't see normal cars with speed past around 130, the engine simply can't take it. The knowledge of the limit of a car engine can be tested using velocity to help ensure a safe driving experience for many.
<math>x(t) = 5 -2t + 6</math>


==See Also==
<math>x(t) = 11 - 2t</math>
[http://www.physicsbook.gatech.edu/Relative_Velocity Relative Velocity]


[http://www.physicsbook.gatech.edu/Speed_and_Velocity Speed and Velocity]
final answer:


[http://www.physicsbook.gatech.edu/Terminal_Speed Terminal Speed]
<math>
  x(t) =
  \begin{cases}
                        -t^2 + 4t + 2 & \text{if $t\leq 3$} \\
                                11-2t & \text{if $t>3$}
  \end{cases}
</math>m


==References==
==Connectedness==


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


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


3. <i>Velocity Expression</i>. Digital image. <i>Physics-Formulas</i>. N.p., n.d. Web. 29 Nov. 2015.
==See Also==


4. <i>Velocity vs Time Graph</i>. Digital image. <i>https://upload.wikimedia.org/wikipedia/commons/a/ae/Velocity-time_graph_example.png</i>. N.p., n.d. Web. 29 Nov. 2015.
*[[Vectors]]
*[[Speed]]
*[[Speed vs Velocity]]
*[[Acceleration]]
*[[Kinematics]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]


==External links==
===External links===


[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]
[http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity The Physics Classroom: Speed and Velocity]
Line 189: Line 243:
[http://hyperphysics.phy-astr.gsu.edu/hbase/vel2.html HyperPhysics: Average 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. <i>Matter and Interactions</i>. Hoboken, NJ: Wiley, 2011. Print.
2. "Velocity." Def. 2. Dictionary.com. N.p., n.d. Web. 29 Nov. 2015.


[[Category:Properties of Matter]]
[[Category:Properties of Matter]]
3. Sas, Wojciech. Angular Velocity Calculator. 02 Nov. 2022.
4. Beck, Kevin. Rotational Motion (Physics): What is it & Why it Matters. 28 Dec. 2020.

Latest revision as of 19:33, 4 December 2022

CLAIMED BY ANJIYA KARIM (FALL 2022)

Main Idea

Velocity is a vector quantity that describes both how fast an object is moving and its direction of motion. It is represented by the symbol [math]\displaystyle{ \vec{v} }[/math] or v, as opposed to [math]\displaystyle{ v }[/math], which denotes 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 momentum and magnetic force, are functions of velocity. Velocity is given in unit distance per unit time. The SI unit for velocity is the meter per second (m/s).

A Mathematical Model

Instantaneous velocity [math]\displaystyle{ \vec{v} }[/math] is defined as:

[math]\displaystyle{ \vec{v} = \frac{d\vec{r}}{dt} }[/math]

where [math]\displaystyle{ \vec{r} }[/math] is a position vector and [math]\displaystyle{ 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]\displaystyle{ \vec{v}_{avg} }[/math]. The average velocity over an interval of time [math]\displaystyle{ \Delta t }[/math] is given by

[math]\displaystyle{ \vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} }[/math]

where [math]\displaystyle{ \Delta \vec{r} }[/math] is the displacement (change in position) over that time interval ([math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \vec{v}_i }[/math] and final velocity [math]\displaystyle{ \vec{v}_f }[/math]:

[math]\displaystyle{ \vec{v}_{avg} = \frac{\vec{v}_i + \vec{v}_f}{2} }[/math]

Unlike the first equation, this equation is only true if 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:

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).







Angular Velocity

Angular velocity, also known as rotational velocity, is how fast a body rotates around a specific center of rotation. It is denoted as omega ω. There are two ways an object can rotate: 

       1) Rotating around its own center of mass. For example, spinning a ball on your finger.
       2) Rotating around a point in space. For example, the rotation of Earth around the Sun.

The higher the angular velocity, the faster the object would rotate. In order to determine the its direction, we can use the right hand rule. By curling the fingers of the right hand in the direction in which the object is turning, the thumb would end up pointing in the direction of the axis of rotation.

Angular Velocity Formulas 

  There are two formulas for calculating angular velocity:
       1) ω=(α2−α1)/t, where α represents angles of a circle in radians and t represents time in seconds.
       2) ω=s/rxt, where s is the arc length, and r is the radius of the circle.
       3) ω=v/r, where v is the linear velocity, and r is the radius.

Calculating Angular Velocity: 

   We can calculate the average angular velocity of the moon in its orbit around the Earth by the following method:
       -The moon takes about 27.3 days (2,358,720 seconds)to orbit around the Earth
       ω=2π/t = 2.7*10^-6/s.

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]\displaystyle{ \vec{v}(t) = \frac{d\vec{r}(t)}{dt} }[/math].

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

[math]\displaystyle{ \vec{a}(t) = \frac{d\vec{v}(t)}{dt} }[/math]

Integral Relationships

Position is the time integral of velocity:

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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.

Click here for velocity simulation

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

Here [1] is another glow script model that represents the change in velocity of the object with respect to time. It also represents how the velocity might affect the change in position of the object.

Velocity Time Graphs

Velocity-Time graphs describe the motion of the objects. Here, the time(s) is often represented at the x-axis since it is the independent variable, while the velocity is represented at the y-axis. The area under these graphs represents the displacement of the object while the slope represents the acceleration. Below is an example of a velocity time graph:

Velocity-Time Graph


Zero Velocity

When a vehicle has zero velocity, it is said to be stationary. Hence, it does not cover any distance per unit time. A graph with a solid line along the x-axis at y=0 is formed.

Constant Velocity

When a vehicle travels at a constant velocity in a particular direction, the acceleration is equal to zero. A graph for such motion consists of a flat line parallel of the x-axis. However, the line is not equal to y=0. In such situations, the vehicle is moving, but the velocity is constant at each time interval.


Increasing/Decreasing Velocity at Constant Rate

When a vehicle starts to travel faster/slower, but the change of velocity is constant per unit time, a graph with a constant positive/negative slope is obtained: Such graph also reflects constant acceleration/deceleration of the object.

Increasing Velocity at a Non-Constant Rate

When a vehicle starts to travel faster, and the change of velocity also increasing per unit time, a curved shaped graph pointing upwards is formed. Such graphs reflect increasing acceleration.

Decreasing Velocity at a Non-Constant Rate

When a vehicle starts to slow down and the change in velocity is also decreasing per unit time, the graph with a curve pointing downwards in obtained. This reflects that the object is decelerating.

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]\displaystyle{ \vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t} }[/math]

[math]\displaystyle{ \vec{v}_{avg} = \frac{\lt -2, 8\gt - \lt 4,-1\gt }{3} }[/math]

[math]\displaystyle{ \vec{v}_{avg} = \frac{\lt -6, 9\gt }{3} }[/math]

[math]\displaystyle{ \vec{v}_{avg} = \lt -2, 3\gt }[/math]m/s

therefore

[math]\displaystyle{ \vec{v}(t=2) = \lt -2, 3\gt }[/math]m/s

Middling

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

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

Solution:

[math]\displaystyle{ \vec{v} = \frac{d \vec{r}}{dt} }[/math]

[math]\displaystyle{ \vec{v} = \lt 12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}\gt }[/math]

Difficult

A particle moves along the x axis. At time t=0, its position is x=2. Its velocity [math]\displaystyle{ v }[/math] varies over time, obeying the following function:

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

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

Solution:

[math]\displaystyle{ x(t) = x(0) + \int_0^t v(t')dt' }[/math] ([math]\displaystyle{ t' }[/math] is a "dummy variable" since [math]\displaystyle{ t }[/math] is already our limit of integration)

for [math]\displaystyle{ t\leq 3 }[/math],

[math]\displaystyle{ x(t) = 2 + \int_0^t 4-2t' dt' }[/math]

[math]\displaystyle{ x(t) = 2 + [4t'-t'^2]_0^t }[/math]

[math]\displaystyle{ x(t) = 2 + 4t - t^2 }[/math]

for [math]\displaystyle{ t\gt 3 }[/math],

[math]\displaystyle{ x(t) = 2 + \int_0^3 4-2t' dt' + \int_3^t -2 dt' }[/math]

[math]\displaystyle{ x(t) = 5 + \int_3^t -2 dt' }[/math]

[math]\displaystyle{ x(t) = 5 + [-2t']_3^t }[/math]

[math]\displaystyle{ x(t) = 5 -2t + 6 }[/math]

[math]\displaystyle{ x(t) = 11 - 2t }[/math]

final answer:

[math]\displaystyle{ x(t) = \begin{cases} -t^2 + 4t + 2 & \text{if $t\leq 3$} \\ 11-2t & \text{if $t\gt 3$} \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, 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

External links

The Physics Classroom: Speed and Velocity

HyperPhysics: Average Velocity

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.

3. Sas, Wojciech. Angular Velocity Calculator. 02 Nov. 2022.

4. Beck, Kevin. Rotational Motion (Physics): What is it & Why it Matters. 28 Dec. 2020.

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