Figure 1 shows water gushing from a large tube through a dam. What is its speed as it emerges? Interestingly, if resistance is negligible, the speed is just what it would be if the water fell a distance

Both

Solving this equation for

We let

where *Torricelli’s theorem*. Note that the result is independent of the velocity’s direction, just as we found when applying conservation of energy to falling objects.

All preceding applications of Bernoulli’s equation involved simplifying conditions, such as constant height or constant pressure. The next example is a more general application of Bernoulli’s equation in which pressure, velocity, and height all change. (See Figure 2.)

### Example 1: **Calculating Pressure: A Fire Hose Nozzle**

Fire hoses used in major structure fires have inside diameters of 6.40 cm. Suppose such a hose carries a flow of 40.0 L/s starting at a gauge pressure of

*Strategy*

Here we must use Bernoulli’s equation to solve for the pressure, since depth is not constant.

*Solution*

Bernoulli’s equation states

where the subscripts 1 and 2 refer to the initial conditions at ground level and the final conditions inside the nozzle, respectively. We must first find the speeds

Similarly, we find

(This rather large speed is helpful in reaching the fire.) Now, taking

Substituting known values yields

*Discussion*

This value is a gauge pressure, since the initial pressure was given as a gauge pressure. Thus the nozzle pressure equals atmospheric pressure, as it must because the water exits into the atmosphere without changes in its conditions.

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