What gauge pressure is required in the city water mains for a stream from a fire hose connected to the mains to reach a vertical height of 13.0 m? (Assume that the mains have a much larger diameter than the fire hose.)

Gauge Pressure = 127.4 KPa

Explanation:

From Bernoulli equation, we know that;

P1 + ρ•g•y1 + (1/2) ρ (v_1) ² = P2 + ρ•g•y2 + (1/2) ρ (v_2) ²

We will set our initial height as 0m and final height as 13m

Also, the question says that the mains have a much larger diameter than the hose. Thus, we can say that v_1 ≈ 0

Furthermore, once the water reaches its altitude of 13m, it stops going higher. Thus, v2 would also be 0.

Now, to simplify the Bernoulli equation further, and given that we are solving for gauge pressure, we can say that the pressure at the top of our column of water is zero, since we need to know the amount of pressure to shoot the water that high in the air anyway. So, P2 = 0

Thus, when we plug in the relevant values into the Bernoulli equation written earlier, we arrive at;

P1 = ρ•g•y2

Density of water = 100 kg/m³

So, P1 = 1000 x 9.8 x 13 = 127,400 Pa = 127.4 KPa