Speed

This page defines and describes speed. Be sure to compare and contrast speed to velocity.

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 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]. Note if the path of the particle is not a straight line, the distance traveled is greater than the magnitude of the displacement ([math]\displaystyle{ \Delta s \gt |\Delta \vec{r}| }[/math]).

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.

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].

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.

Examples

Simple

A runner runs a 560m lap counterclockwise around a circular track in 200s.

a) What is the average speed of the runner over the course of the lap?

b) What is the average velocity of the runner over the course of the lap?

Solutions:

a) To find the average speed of the runner, let us use the following formula:

[math]\displaystyle{ |v|_{avg} = \frac{d}{\Delta t} }[/math]

[math]\displaystyle{ |v|_{avg} = \frac{560}{200} }[/math]

[math]\displaystyle{ |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]\displaystyle{ \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]\displaystyle{ \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]\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 Kinetic Energy 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 velocity of moving parts is important in virtually every type of machinery.

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. Velocity Expression. Digital image. Physics-Formulas. N.p., n.d. Web. 29 Nov. 2015.

4. Velocity vs Time Graph. Digital image. https://upload.wikimedia.org/wikipedia/commons/a/ae/Velocity-time_graph_example.png. N.p., n.d. Web. 29 Nov. 2015.