Question 9.13: Shoot-Out at the Old Water Tank GOAL Apply Bernoulli’s equat...
Shoot-Out at the Old Water Tank
GOAL Apply Bernoulli’s equation to find the speed of a fluid.
PROBLEM A nearsighted sheriff fires at a cattle rustler with his trusty six-shooter. Fortunately for the rustler, the bullet misses him and penetrates the town water tank, causing a leak (Fig. 9.32 on page 304). (a) If the top of the tank is open to the atmosphere, determine the speed at which the water leaves the hole when the water level is 0.500 m above the hole. (b) Where does the stream hit the ground if the hole is 3.00 m above the ground?
STRATEGY (a) Assume the tank’s cross-sectional area is large compared to the hole’s (A_{2}\gt \gt A_{1}) , so the water level drops very slowly and v_{2}\approx0. Apply Bernoulli’s equation to points ① and ② in Figure 9.31, noting that {\mathbf{}}P_{1} equals atmospheric pressure {\mathbf{}}P_{o} at the hole and is approximately the same at the top of the water tank. Part (b) can be solved with kinematics, just as if the water were a ball thrown horizontally.
(a) Find the speed of the water leaving the hole.
Substitute P_{1}=P_{2}=P_{0} and v_{2}\approx0 into Bernoulli’s equation, and solve for v_{1}:
P_{0}+{\frac{1}{2}}\rho v_{1}^{2}+\rho g y_{1}=P_{0}+\rho g y_{2}
v_{1}={\sqrt{2g(y_{2}-y_{1})}}={\sqrt{2g h}}
v_{1}={\sqrt{2(9.80~{\mathrm{m/s}}^{2})(0.500~{\mathrm{m}})}}=3.13~{\mathrm{m/s}}
(b) Find where the stream hits the ground.
Use the displacement equation to find the time of the fall, noting that the stream is initially horizontal, so v_{0y}~=~0.
\Delta y=-\textstyle{\frac{1}{2}}g t^{2}+v_{0y}t
-3.00\mathrm{~m}=-(4.90\mathrm{~m}/s^{2})t^{2}
t=0.782\ \mathrm{s}
Compute the horizontal distance the stream travels in this time:
x=v_{0x}t=(3.13~{\mathrm{m/s}})(0.782~{\mathrm{s}})=~\textstyle{{2.45~\mathrm{m}}}
REMARKS As the analysis of part (a) shows, the speed of the water emerging from the hole is equal to the speed acquired by an object falling freely through the vertical distance h. This is known as Torricelli’s law.