Refill the Pool

Suppose you noticed the water level of your neighbor's swimming pool to be about 6 inches low. You inform him of the fact whereupon he directs a garden hose into the pool and turns it on. The pool has vertical sides and measures 15 by 36 feet.

The sketch shows the geometry of this "pool filling" event. For a moment the two of you watch the water flowing into the pool. Then, knowing you are an engineer, the neighbor asks: "When should I return and turn the water off?"

♦ Step One: First we use the geometry of the flow and its conditions to solve the momentum equation for the velocity of the water issuing from the hose. We take the system to be a small droplet or amount of the that issues from the hose then free-falls to the water. The general geometry with coordinates, the initial, and final positions of the water are shown to the right.

We take an amount of water exiting the nozzle. With the nozzle exit as origin, Newton's 2nd Law applies with gravity as the acting force:

(1)

Eqn. 1 is called the governing equation. Two conditions that apply are the position at start (2) and the position of entry into the water (3).

(2)

(3)

Careful solution of the differental equation (with conditions) proceeds as follows.

Step Two: Our next task is to define a system and apply the mass equation. The sketch shows the pool before and after the fill.

The system is the water in the pool which has the initial depth, H and the final depth is H + 0.5 ft. Solution of the mass equation is as follows: