What Is Speed?

Anyone who drives a car is familiar with the concept of speed. It's the number on the speedometer that tells you how fast the car is going. In automobiles, speed is measured in miles per hour (mph) or kilometers per hour (kph), but the units physicists most often use for speed are meters per second (m/s). You can get an accurate definition of speed simply by examining these units. They are units of distance traveled (d) divided by units of time (t), and that is basically the definition of speed (S). In the language of mathematics you write this as

\(S=\frac{d}{t}\)

The speedometer in your car tells you how far you go in an hour if you maintain a constant speed, but drivers seldom do that. In a typical hour of driving, the speed recorded by the speedometer changes constantly, and the actual distance you travel in an hour is an average of all these speeds. The total distance you travel in a unit of time, such as an hour, is your average speed (S_av):

\(S_{av}=\frac{\text{total distance}}{\text{ total time}}\)

The speed displayed on the speedometer is your instantaneous speed (S_I). Physicists express instantaneous speed by defining it as the change in position (x) between two time intervals, t₁ and t₂ and letting the time interval approach zero.

\(S_1 = \frac{x_2-x_1}{t_2-t_1}=\frac{\Delta x}{\Delta t}\)

If you let ∆t approach zero, you get a mathematical expression known as a derivative, which by convention is written dx/dt. For physicists, the most accurate expression for instantaneous speed is

\(S_1=\frac{dx}{dt}\)

People often use the words speed and velocity interchangeably, but they don't mean the same thing. Velocity is a vector quantity, which means it has a directional component, whereas speed is a scalar quantity which does not take direction into account.

To see why direction is important, consider the difference in time it takes to drive between two points on a straight road and a winding one. If the road is straight, you can average all the instantaneous speeds recorded by the speedometer and get the same average speed as you would by dividing the total distance by the total time. If the road is windy, however, these two numbers will be different. This is because the directional component of the velocity toward your destination decreases every time the road veers to the left or right.

Velocity is sometimes denoted by the letter v with an arrow over it to signify that it's a vector quantity, but the arrow isn't really necessary. By definition, velocity has a directional component.

When an object is rotating, the rotational speed is the number of complete revolutions it makes in a unit time. The most common units are revolutions per minute (rpm). Points in a spinning disk have a forward velocity that is constantly changing direction. The tangential speed is the forward velocity at any given moment, and it's affected by the radial distance from the center of rotation. Points farther from the center are moving faster than points closer to the center. You calculate the tangential speed of a point on a rotating disk using the expression:

\(S_{tangential}=\text{rotational speed}\times \text{ radial distance}\)