Question

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. Find an algebraic expression for the flow speed v with which the water exits the hole at height y.

Accepted Answer

Step 1: Begin by understanding the physical principles involved. The flow speed of water exiting the hole is determined by Torricelli's theorem, which is derived from Bernoulli's equation. Torricelli's theorem states that the speed of efflux, v, is given by $$ v = \sqrt{2 g (h - y)} $$, where $$ g  is $$the acceleration due to gravity, $$ h  is $$the total height of the water column, and $$ y  is $$the height of the hole from the ground. Step 2: Apply Bernoulli's equation to the system. Bernoulli's equation states that the total mechanical energy (pressure energy, kinetic energy, and potential energy) is conserved along a streamline. At the surface of the water (height $$ h $$), the velocity is approximately zero, and the pressure is atmospheric. At the hole (height $$ y $$), the pressure is also atmospheric, but the water has a velocity $$ v . $$Using this, derive the expression $$ v = \sqrt{2 g (h - y)} . $$Step 3: Recognize that the term $$ h - y $$ represents the vertical distance between the water surface and the hole. This distance determines the potential energy difference that is converted into kinetic energy as the water exits the hole. Step 4: Note that the flow speed $$ v $$ depends only on the height difference $$ h - y $$ and the gravitational acceleration $$ g . $$This means that the flow speed is independent of the size of the hole or the rate at which water is replenished, as long as $$ h $$ remains constant. Step 5: Conclude that the algebraic expression for the flow speed $$ v  is$$ $$ v = \sqrt{2 g (h - y)} . $$This formula can now be used to analyze the motion of the water as it exits the hole and to determine other quantities, such as the range of the water.

Verified video answer for a similar problem:

Key Concepts

Bernoulli's Principle

Torricelli's Law

Projectile Motion