Swinging Over the Water Hole

A child takes a running start, grabs hold of the free-hanging end of a rope at the edge of a water hole at x = 0, swings upward, and lets go of the rope at some point, flying freely through the air until he lands in the water at x = R . For small initial speeds, the maximum range R is obtained by releasing the rope just short of its turning point and dropping almost straight down into the water from rest. On the other hand, as a child's running speed gets larger and larger (compared to the square root of the product of the length of the rope and earth's gravitational field), the rope should be released at an angle of about pie / 4 to maximize the range. That is, the optimum trajectory of a child is dominated by pendulum motion at low running speeds and by projectile motion at high initial speeds.
Published in Lat. Am. J. Phys. Educ., v5 n2 p335-337, June 2011.