Wednesday, July 9, 2008

Playing Catch in a Rotating Spaceship

It's a standard physics problem, but you don't really get it until you work through it for yourself.

You're standing in a circular spaceship, which is rotating to provide artificial gravity. You throw a ball straight up in the air. What path do you see it follow?

The longish version of the answer is here.

What was interesting was that as I went along, more and more little things fell out of the math. Once I had the equations for the trajectory of the ball, for example, I saw that I could take a Taylor expansion about any point in the trajectory, and the Coriolis and centripetal accelerations would fall right out - even though I made no reference to them in the solution (which is purely kinematical).

It also turns out that the shape of the trajectory is specified completely by the angular velocity of the ship's rotation and the ratio of the velocity of the ball to the size of the ship. If you make the ship twice as big, and throw the ball twice as hard, the trajectory will have exactly the same shape.

I also made a Mathematica Demonstration plotting the trajectory as a function of angular velocity, size of ship, and how hard you throw the ball. If they publish it it'll be online soon, but in the mean time there are some sample plots on the pdf.

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