# Is there an optimum length for a swing for a childs mass?

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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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