Chapter 14, Problem 68EAP

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 $y\to 0$ because the water squirts right on to the ground. The range also approaches zero as $y\to h$ 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 $x_{\text{max}}$ and the height y of the hole. "Real" water won't achieve quite this range because of viscosity, but it will be close.