Randall D. Knight — Fluids and Elasticity: Problem Problem_64

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 \( y \rightarrow 0 \) because the water squirts right onto the ground. The range also approaches zero as \( y \rightarrow h \) 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.