Tumble: Bouncing Balls

Kidinu: Hey, what happened? All those lines, and only one ball?

Ahmes [looking over Kidinu's shoulder]: I thought it would help explain things.

Kidinu: I beg your pardon, but I thought I was controling this computer.

Ahmes: Please excuse my intrusion, but I think I can explain the physics, if you would like.

Katyayana: I'd like to hear that.

Ahmes: Let us start with one ball. It's simpler that way. Notice how it keeps going at the same rate in the same direction until it hits a wall.

Katyayana: Well, yes, but the other balls kept falling down. Why isn't this one?

Ahmes: I thought it would be easier if we started without gravity. We can add it later. We can specify where the ball is by using coordinates. The coordinates of the center of the ball are a pair of numbers (x,y). The number x indicates how far right (if x is positive) or how far left (if x is negative) it is from the vertical center line, while the number y indicates how far above (if y is positive) or how far below (if y is negative) from the horizontal center line.

Kidinu: Yes, of course, we know all about coordinates. You don't have to tell us about them.

Ahmes: Well, all those lines you asked about earlier are just the grid lines for the coordinate system.

Katyayana: What about motion?

Ahmes: I was just getting to that. The velocity of the ball can also be given in terms of coordinates.

Kidinu: I thought velocity was just how fast something was going. You don't need two numbers for that.

Ahmes: Well, no, if you just want to know how fast a ball is moving, you only need one number, but if you want velocity to include the direction the ball is going, you'll need more information. You can take velocity to be a pair (x',y') of numbers. The number x' indicates how the x-coordinate is changing. So if the ball is moving right, then x' is positive. Likewise number x' indicates how the y-coordinate is changing. So if the ball is moving upward, then y' is positive.

Katyayana: I get it. If it's moving very fast directly right, then x' is big, but y' is 0. But if it's moving on a diagonal up to the right, then x' would equal y'.

Kidinu: Why was it, now, that you brought up coordinates?

Ahmes: As I said before, a ball continues to move at the same rate in the same direction until it hits something. That means its velocity (x',y') is constant during that period.

to the intro. to the cover page. to wall collisions.

David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610