A water tank of height h has a small hole at height y. The water is replenished to keep h from changing. The water squirting from the hole has range x. The range approaches zero as y → 0 because the water squirts right onto the ground. The range also approaches zero as y → h because the horizontal velocity becomes zero.
Thus there must be some height y between 0 and h for which the range is a maximum.
a. Find an algebraic expression for the flow speed v with which the water exits the hole at height y.
b. Find an algebraic expression for the range of a particle shot horizontally from height y with speed v.
c. Combine your expressions from parts a and b. Then find the maximum range x_max and the height y of the hole. “Real” water won’t achieve quite this range because of viscosity, but it will be close.