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.