The science museum that I volunteer at recently opened up the traveling math exhibit that I’ve been looking forward to for a while. One of the activities features a piece of equipment that, as a physics student/teacher, is near and dear to my heart: the motion sensor. As you walk towards and away from the sensor, it records how far away you are and plots this distance on a graph.

It’s a nifty setup that gets people thinking about how motion can be described both visually and mathematically, but it does raise the question: how does the sensor *know* how far away you are?

The explanation is best illustrated with a game of catch.

Imagine that you and your friend are both capable of throwing a ball at exactly 10 m/s. Imagine that you also have an incredibly accurate stopwatch that you can operate while simultaneously throwing or catching a ball.

Your friend is standing an unknown distance away from you. You throw the ball at her and start the stopwatch; as soon as she catches the ball, she throws it back at you at exactly the same speed. As soon as you catch the ball, you stop the stopwatch. Since you know the speed of the ball and the amount of time required for it to make the round trip, you can calculate the distance between the two of you:

• speed = total distance/total time

(Technically, it’s *average* speed = total distance/total time, but in this situation the ball is moving at a constant speed. The ‘average’ would come into play if the ball was speeding up or slowing down.)

Suppose the stopwatch reads exactly 4 seconds when the ball returns to you. At our given speed of 10 m/s, we can calculate how far the ball traveled.

• 10 m/s = total distance/4 seconds

The ball covered a distance of 40 m. An important detail to remember here is that this is the *round trip* distance. Since the ball had to leave your hands, get to your friend, and come back to you, your friend is standing *20* m away from you.

The motion detector uses a similar process to determine how far away you are; instead of using a ball, it uses sound. The device sends out a constant stream of clicking noises, each of which bounces off of you and returns to the detector. (While we can’t hear it ourselves in this case, this bounced clicking noise, like all reflected sound waves, is an echo.) Given that the speed of sound in air is fairly constant, the detector’s software can use this speed and the amount of time that passes between an emitted click and the return of its echo to calculate the distance. Making this calculation several times every second results in a smooth graph of your motion.

I’m going to stop now before I go on for pages about motion graph analysis (and if you don’t believe that I can, you clearly haven’t talked to my physics students). That will be a post (or several) for another day.