**Problem statement:**

A fountain shoots a vertical jet of water to a maximum height H = 25 m. The fountain has a d = 5 cm nozzle at ground level. The water pump is h = 3 m below the ground (see figure). If the pipe to the nozzle has a diameter D = 8 cm, what pressure must the pump of the fountain supply?

__Givens__: ρ = 10^{3} kg/m^{3}; p_{0} = 10^{5} Pa; g = 10 m/s^{2}

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**Solution:**

Let’s assume a **steady flow**. In this case we can use the continuity equation and Bernoulli’s equation to solve this problem.

In order to apply Bernoulli’s equation we must find the speed of water v_{2} when it exists the pump.

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Now, since we know the diameters of both pipes we can apply the continuity equation to states 1 and 2:

And isolating v_{2}:

Both diameters are givens of this problem. We can also determine the water speed in state 1, v_{1}, because we know the height H it reaches. At this point its speed will be zero Its acceleration is that of gravity, g, so:

Now we make use of **Bernoulli’s equation** between states 1 and 2. We take h = 0 at state 2. At the open end 1 of the pipe the pressure is atmospheric pressure p_{0}.

Bernoulli’s equation is:

Isolating p_{2} and substituting:

Finally, substituting v_{1} as a function of H as well as the givens of the problem we obtain:

**Make sure that you include units in your results.**