# Speed

**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.