Speed: Difference between revisions

From Physics Book
Jump to navigation Jump to search
No edit summary
mNo edit summary
 
(29 intermediate revisions by one other user not shown)
Line 1: Line 1:
This page defines and describes speed. Be sure to compare and contrast speed to [[Velocity|velocity]].
'''Nina Casselberry Fall 2021'''
 
This page defines and describes speed. Be sure to distinguish between speed and [[Velocity|velocity]]. See [[Speed vs Velocity]] for similarities and differences.


==Main Idea==
==Main Idea==


Speed is a scalar quantity that describes how fast an object is moving. It is represented by the symbol <i>v</i>, as opposed to <math>\vec{v}</math> and <b>v</i>, which denote [[Velocity|velocity]]. 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|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 Units|SI unit]] for speed is the meter per second (m/s).
Speed is a scalar quantity that describes how fast an object is moving. It is represented by the symbol <math>v</math>, as opposed to <math>\vec{v}</math> and <b>v</b>, which denote [[Velocity|velocity]]. Speed is defined as the magnitude of the rate of change of position with respect to time. In calculus terms, it is the time derivative of distance traveled. Speed is the magnitude of velocity. Speed is an instantaneous value, so it may change over time. Several properties in physics, such as [[Kinetic Energy|kinetic energy]], are functions of speed. Speed is given in unit distance per unit time. The [[SI Units|SI unit]] for speed is the meter per second (m/s).


===A Mathematical Model===
===A Mathematical Model===


Instantaneous velocity <math>\vec{v}</math> is defined as:
Instantaneous speed <math>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====
<math>v = \frac{ds}{dt}</math>


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 acceleration over an interval of time <math>\Delta t</math> is given by
where <math>s</math> is distance traveled and <math>t</math> is time.


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


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


====Derivative Relationships====
<math>v_{avg} = \frac{\Delta s}{\Delta t}</math>


Velocity is the time derivative of position:
where <math>\Delta s</math> is the distance traveled during the time interval <math>\Delta t</math>.


<math>\vec{v}(t) = \frac{d\vec{r}(t)}{dt}</math>.
Average speed is often confused with average velocity. A section on the [[Speed vs Velocity]] page goes into depth about the difference.


[[Acceleration]], in turn, is the time derivative of velocity:
As the duration of the time interval <math>\Delta t</math> becomes very small, the value of the average speed over that time interval approaches the value of the instantaneous speed at any point in time within that interval.


<math>\vec{a}(t) = \frac{d\vec{v}(t)}{dt}</math>
If speed is constant over some time interval, the average speed over that time interval equals that constant speed.


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


Position is the time integral of velocity:
Distance traveled is the time integral of speed:


<math>\vec{r}(t) =  \int \vec{v}(t) \ dt</math>.
<math>s(t) =  \int 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:
Integrating the speed function as a definite integral from one time coordinate to another yields the distance traveled over that time interval:


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


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


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


====Kinematic Equations====
A runner runs at a circular track with radius 100m. The runner runs 3 laps in 250s at a constant speed. What is the speed of the runner?


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.
Solution:


====In Physics====
The runner moved at a constant speed, so that speed must also be the average speed the over the time interval 250s time interval over which the laps were completed. Let us therefore use the following formula:


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>v_{avg} = \frac{\Delta s}{\Delta t}</math>


===A Computational Model===
<math>v_{avg} = \frac{2\pi*100}{250}</math>


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>v_{avg} = 2.51</math>m/s


[https://www.glowscript.org/#/user/YorickAndeweg/folder/PhysicsBookFolder/program/Velocity Click here for velocity simulation]
The speed of the runner was 2.51m/s.


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


==Examples==
On graduation day, you excitedly throw your cap into the air! The cap leaves your hands when they are fully stretched upwards at a height of 2 meters above the ground. The cap then reaches a maximum height of 10 meters above the ground before falling straight back down to your hands, which have not moved since you threw the cap. If the cap was in the air for a total of four seconds, what was the average speed of the cap during that period? Your cap is light and therefore affected by air resistance, so do not assume that acceleration is constant.


===Simple===
Solution:


A runner runs a 560m lap counterclockwise around a circular track in 200s.
<math>v_{avg} = \frac{\Delta s}{\Delta t}</math>


a) What is the average speed of the runner over the course of the lap?
On its way up, the cap traveled from a height of 2m to a height of 10m, which is a distance of 8m. On its way down, the cap traveled from a height of 10m to a height of 2m, which is a distance of 8m. The total distance traveled by the cap in the 4 second time interval is therefore 8+8=16m.


b) What is the average velocity of the runner over the course of the lap?
<math>v_{avg} = \frac{16}{4}</math>


Solutions:
<math>v_{avg} = 4</math>m/s.


a) To find the average speed of the runner, let us use the following formula:
===Difficult===
 
<math>|v|_{avg} = \frac{d}{\Delta t}</math>
 
<math>|v|_{avg} = \frac{560}{200}</math>
 
<math>|v|_{avg} = 2.8</math>m/s
 
b) To find the average value of the velocity vector over time, let us use the following formula:
 
<math>\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}</math>.
 
Because the runner ran a single lap and ended at the point in space where they began, their displacement <math>\Delta \vec{r}</math> is <0,0>m! The runner's average velocity was therefore <0,0>m/s. This makes sense because the velocity vector spends time pointing in every possible direction in the plane of the track, so each instantaneous velocity at any point along the track cancels with other instantaneous velocities at other points along the track.
 
===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>
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>
Line 94: Line 78:


Solution:
Solution:
Let us first find the velocity of the particle as a function of time. From there, the speed of the particle can be found by taking its magnitude.


<math>\vec{v} = \frac{d \vec{r}}{dt}</math>
<math>\vec{v} = \frac{d \vec{r}}{dt}</math>
Line 99: Line 85:
<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>
<math>\vec{v} = <12 \pi \cos(2 \pi t), -18t^2, 8e^{8t}></math>


===Difficult===
<math>v = |\vec{v}| = \sqrt{(12 \pi \cos(2 \pi t))^2 + (-18t^2)^2 + (8e^{8t})^2}</math>
 
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>
<math>v = \sqrt{144 \pi^2 \cos^2(2 \pi t) + 324t^4 + 64e^{16t}}</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==
==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 [[Kinetic Energy]] depend on the velocities of objects.
Since the goal of the field of mechanics is to predict the motion of systems, speed is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as [[Kinetic Energy]] depend on the speeds 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 velocity of moving parts is important in virtually every type of machinery.
The use of speed 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 speeds of moving parts are important in virtually every type of machinery.


==See Also==
==See Also==


[[Speed]]
*[[Velocity]]
 
*[[Speed vs Velocity]]
[[Acceleration]]
 
[[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]
[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==
==References==


1. Chabay, Ruth W., and Bruce A. Sherwood. <i>Matter and Interactions</i>. Hoboken, NJ: Wiley, 2011. Print.
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.
3. <i>Velocity Expression</i>. Digital image. <i>Physics-Formulas</i>. N.p., n.d. Web. 29 Nov. 2015.
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.


[[Category:Properties of Matter]]
[[Category:Properties of Matter]]

Latest revision as of 00:25, 17 November 2021

Nina Casselberry Fall 2021

This page defines and describes speed. Be sure to distinguish between speed and velocity. See Speed vs Velocity for similarities and differences.

Main Idea

Speed is a scalar quantity that describes how fast an object is moving. It is represented by the symbol [math]\displaystyle{ v }[/math], as opposed to [math]\displaystyle{ \vec{v} }[/math] and v, which denote velocity. Speed is defined as the magnitude of the rate of change of position with respect to time. In calculus terms, it is the time derivative of distance traveled. Speed is the magnitude of velocity. Speed is an instantaneous value, so it may change over time. Several properties in physics, such as kinetic energy, are functions of speed. Speed is given in unit distance per unit time. The SI unit for speed is the meter per second (m/s).

A Mathematical Model

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

[math]\displaystyle{ v = \frac{ds}{dt} }[/math]

where [math]\displaystyle{ s }[/math] is distance traveled and [math]\displaystyle{ t }[/math] is time.

Average Speed

Since speed is an instantaneous quantity, it can change over time. Over any given time interval, there is an average speed value denoted [math]\displaystyle{ v_{avg} }[/math]. The average speed over an interval of time [math]\displaystyle{ \Delta t }[/math] is given by

[math]\displaystyle{ v_{avg} = \frac{\Delta s}{\Delta t} }[/math]

where [math]\displaystyle{ \Delta s }[/math] is the distance traveled during the time interval [math]\displaystyle{ \Delta t }[/math].

Average speed is often confused with average velocity. 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 speed over that time interval approaches the value of the instantaneous speed at any point in time within that interval.

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

Integral Relationships

Distance traveled is the time integral of speed:

[math]\displaystyle{ s(t) = \int v(t) \ dt }[/math].

Integrating the speed function as a definite integral from one time coordinate to another yields the distance traveled over that time interval:

[math]\displaystyle{ \Delta s_{t_1,t_2} = \int_{t_1}^{t_2} v(t) \ dt }[/math].

Examples

Simple

A runner runs at a circular track with radius 100m. The runner runs 3 laps in 250s at a constant speed. What is the speed of the runner?

Solution:

The runner moved at a constant speed, so that speed must also be the average speed the over the time interval 250s time interval over which the laps were completed. Let us therefore use the following formula:

[math]\displaystyle{ v_{avg} = \frac{\Delta s}{\Delta t} }[/math]

[math]\displaystyle{ v_{avg} = \frac{2\pi*100}{250} }[/math]

[math]\displaystyle{ v_{avg} = 2.51 }[/math]m/s

The speed of the runner was 2.51m/s.

Middling

On graduation day, you excitedly throw your cap into the air! The cap leaves your hands when they are fully stretched upwards at a height of 2 meters above the ground. The cap then reaches a maximum height of 10 meters above the ground before falling straight back down to your hands, which have not moved since you threw the cap. If the cap was in the air for a total of four seconds, what was the average speed of the cap during that period? Your cap is light and therefore affected by air resistance, so do not assume that acceleration is constant.

Solution:

[math]\displaystyle{ v_{avg} = \frac{\Delta s}{\Delta t} }[/math]

On its way up, the cap traveled from a height of 2m to a height of 10m, which is a distance of 8m. On its way down, the cap traveled from a height of 10m to a height of 2m, which is a distance of 8m. The total distance traveled by the cap in the 4 second time interval is therefore 8+8=16m.

[math]\displaystyle{ v_{avg} = \frac{16}{4} }[/math]

[math]\displaystyle{ v_{avg} = 4 }[/math]m/s.

Difficult

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:

Let us first find the velocity of the particle as a function of time. From there, the speed of the particle can be found by taking its magnitude.

[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]

[math]\displaystyle{ v = |\vec{v}| = \sqrt{(12 \pi \cos(2 \pi t))^2 + (-18t^2)^2 + (8e^{8t})^2} }[/math]

[math]\displaystyle{ v = \sqrt{144 \pi^2 \cos^2(2 \pi t) + 324t^4 + 64e^{16t}} }[/math]

Connectedness

Since the goal of the field of mechanics is to predict the motion of systems, speed is critical to this branch of physics. In addition to being a value of interest in and of itself, properties such as Kinetic Energy depend on the speeds of objects.

The use of speed 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 speeds of moving parts are important in virtually every type of machinery.

See Also

External links

The Physics Classroom: Speed and Velocity

References

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

Retrieved from "http://www.physicsbook.gatech.edu/index.php?title=Speed&oldid=39167"