Rhett Allain

If the internet was your online physics textbook, the Slow Mo Guys would be writing a good number of those homework questions. Of course the Slow Mo Guys are Gavin and Dan, and they use a super-high-speed camera and look at stuff around us. OK—in this case, they aren't looking at normal things. They are looking at the motion of a high-speed shell fired from a tank.

But how do you see this tank shell if it is traveling around 2,000 feet per second (609 m/s)? You could just put the camera far enough away that the path of the projectile would be in the frame. However, in that case you would barely see the fast-moving object. It would be too small in the video. OK, what about getting close to the path so that the shell looks larger? Yes, you would see it, but just for a tiny fraction of the total path.

The solution to this problem is to use both methods. Get the camera close to the path and then rotate the view as the projectile passes. The Slow Mo Guys are going to put the camera 82 feet from the path, which means it would have to rotate at around 3,000 degrees per second—which is pretty much impossible. But instead of rotating a camera, they will just use a rotating mirror. The camera looks at the mirror and from that it can see the tank shell. Then the camera can stay in place while the mirror rotates. Perfect.

Here is the real question. How do you determine the angular position of a mirror so that it can track the projectile? The answer: polar coordinates. Yes, you thought they were just joking when they made you do stuff with polar coordinates in math class. Surprise. You actually need this sometimes.

Let's do this. How about a review of two different coordinate systems: Cartesian and polar coordinates. Also, how do you use polar coordinates to track a super fast projectile?

Cartesian Coordinates

This is the one you are likely familiar with. In two dimensions, there is an x-axis and a y-axis. They are perpendicular to each other. Once you pick the origin, you can describe the location of an object with x and y coordinates.