Optimal angle on a swing

This is another interesting problem that is similar to– but slightly trickier than– the golf problem from last year.

Problem: It’s recess, and you are swinging on a playground swing set, and because it’s against playground rules and girls are watching, you want to jump off the swing.  How should you time your release (i.e., at what angle from vertical should you let go) to maximize the distance traveled before hitting the ground?

I like this problem, because there is a lot of freedom to make simplifying assumptions or not, yielding solutions ranging from complex and only numerically tractable to simpler, more elegant, and still reasonably accurate.  For example, let’s make it easier by putting the swing set on a flat moon (i.e., gravity acts directly downward with no air resistance), and you are a spherical chicken of negligible radius (i.e., a point mass) riding the swing whose seat just misses the ground when it hangs vertical, and you want to maximize the over-the-ground distance measured from that point at which the swing hangs vertical.

 

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