Instantaneous velocity: Velocity at a specific moment.

Objects can speed up or slow down, or they can change direction. In other words, their velocity can change. For example, if you drop an egg off a 40-story building, the egg’s velocity will change: It will move faster as it falls. Someone on the building’s 39^th floor would see it pass by with a different velocity than would someone on the 30^th.

When we use the word “instantaneous,” we describe an object’s velocity at a particular instant. In Concept 1, you see a snapshot of a toy mouse car at an instant when it has a velocity of positive six meters per second.

The fable of the tortoise and the hare provides a classic example of instantaneous versus average velocity. As you may recall, the hare seemed faster because it could achieve a greater instantaneous velocity than could the tortoise. But the hare’s long naps meant that its average velocity was less than that of the tortoise, so the tortoise won the race.

When the average velocity of an object is measured over a very short elapsed time, the result is close to the instantaneous velocity. The shorter the elapsed time, the closer the average and instantaneous velocities. Imagine the egg falling past the 39^th floor window in the example we mentioned earlier, and let’s say you wanted to determine its instantaneous velocity at the midpoint of the window.

You could use a stopwatch to time how long it takes the egg to travel from the top to the bottom of the window. If you then divided the height of the window by the elapsed time, the result would be close to the instantaneous velocity. However, if you measured the time for the egg to fall from 10 centimeters above the window’s midpoint to 10 centimeters below, and used that displacement and elapsed time, the result would be even closer to the instantaneous velocity at the window’s midpoint. As you repeated this process "to the limit" − measuring shorter and shorter distances and elapsed times (perhaps using motion sensors to provide precise values) − you would get values closer and closer to the instantaneous velocity.

To describe instantaneous velocity mathematically, we use the terminology shown in Equation 1. The arrow and the word “lim” mean the limit as Δt approaches zero. The limit is the value approached by the calculation as it is performed for smaller and smaller intervals of time.

The limit equation defines a derivative, as shown in Equation 2. Instantaneous velocity is the derivative of position with respect to time. This means that if you have an expression representing the position of an object as a function of time, differentiating it gives you the velocity as a function of time.

To give you a sense of velocity and how it changes, let’s again use the example of the egg. We calculate the velocity at various times using an equation you may have not yet encountered, so we will just tell you the results. Let’s assume each floor of the building is four meters (13 ft) high and that the egg is being dropped in a vacuum, so we do not have to worry about air resistance slowing it down.

One second after being dropped, the egg will be traveling at 9.8 meters per second; at three seconds, it will be traveling at 29 m/s; at five seconds, 49 m/s (or 32 ft/s, 96 ft/s and 160 ft/s, respectively.)

After seven seconds, the egg has an instantaneous velocity of 0 m/s. Why? The egg hit the ground at about 5.7 seconds and therefore is not moving. (We assume the egg does not rebound, which is a reasonable assumption with an egg.)

Physicists usually mean “instantaneous velocity” when they say “velocity” because instantaneous velocity is often more useful than average velocity. Typically, this is expressed in statements like “the velocity when the elapsed time equals three seconds.”