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If you want to maximize the maximum velocity a child could go, what would be the optimum height?

If you wanted to maximize the efficiency of a child "pumping" their legs to gain velocity, what would be the optimum height?

I hope that I am correctly understanding how a child pumping their legs makes it possible for them to gain momentum. They shift the center of mass of the swing+child outside the normal position of being in-line with the rope/chain. By shifting the center of mass behind t's normal position, they are allowing gravity to place a small amount of force in their direction of motion. When their center of mass and the rope is parallel with the force of gravity (like a plumb bob), they cannot impart any additional velocity on their swing motion. When they are at the apex, they can impart the most force.

I imagine too short a swing, and the child will not have a very high maximum velocity, and the time between pumps will be too short.

I imagine too long, and the wind resistance will be too great, and the childs mass compared to the entire swing will be too little.

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The first part (maximize peak velocity) is simple: the longer the swing the better. Ultimately the maximum velocity depends on the maximum height the swinger can obtain, and as they are limited to a bit less than 90 degrees from the vertical that is a bit less than the length of the swing.

The second part is not trivial. Indeed I am not even sure where to start defining "efficiency of pumping".

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Yeah, the second half would involve a double pendulum type system, except that second pendulum swings abnormally (as in, it does not swing in the manner that a double pendulum does). Ick. –  Manishearth Feb 6 '12 at 3:13
Yes, because velocity is a function of gravity, longer swings are better. There still must be some maximum length, at which the mass of the child is insignificant to the weight of the rope, chain, seat. At some point, the child would be unable to overcome the inertia of the swing to get it going(and wind resistance). This is kind of what I meant by pump efficiency. A mile high swing that weighed a ton, couldn't begin to get any height using the mass of a child's leg pumps. –  user1873 Feb 6 '12 at 14:39
Well, I suspect that $M_{swing} \gg M_{legs}$ is pretty bad for the efficiency in that meaning. With a child's legs being less than 10 kg, that suggests that swing massing over ~100 kg would be impossible to move much. Of course 100 kg of steel cable makes one heck of a swing. I suspect that air resistance limits the height before we get that long, which makes my answer incomplete. In my defense I assumed you were talking about reasonably realizable swings. The tallest one I've used was about 6 meters, and I was able to get it near horizontal. That was fast. –  dmckee Feb 6 '12 at 19:00
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