How can a car have a speed at a single instant?

A car has a speed at a single instant because that speed is defined as the value the average speeds settle towards as you shrink the time window. A speedometer does not have to catch the car in one frozen moment. It is a gauge that measures and displays the instantaneous speed of a vehicle, and the idea behind that word is what calculus was built to make precise.

Start with the speed you already know. Average velocity is the constant velocity that would take you the same distance in the same time as your real, stop-and-go trip. Drive for an hour and you can divide the distance by the hour. But that number hides everything that happened inside the hour. To ask how fast you were going at one moment, you can shrink the window: use the last ten minutes, then the last minute, then the last second. Each window gives a different average, and they close in on one number. The instantaneous velocity is the limit of the average velocity as the time interval approaches zero.

Real instruments quietly do this. One early kind of speedometer measured the distance travelled over a precise short interval of time, in some models three quarters of a second. It is an average over a tiny window, treated as the speed at that moment. Another way to picture the result: the instantaneous velocity is the velocity the object would carry on with if it stopped speeding up or slowing down at that instant.

The idea sounds slippery because a single instant has no duration, so no distance can be covered in it. That puzzle is old. The Greek philosopher Zeno argued that an arrow in flight is, at any one instant, neither moving to where it is nor to where it is not. Seen as a limit of ever smaller windows, speed at an instant stops being a paradox and becomes a well-defined number.