Answer to Question #184390 in Mechanics | Relativity for Konoha

Question

A container of water has a cross-sectional area of 0.1m². A piston sits on top of the water (see the following figure). There is a spout located 0.15 m from the bottom of the tank, open to the atmosphere, and a stream of water exits the spout. The cross-sectional area of the spout is 7.0×10⁻⁴ m²

1. What is the velocity of the water as it leaves the spout?
2. If the opening of the spout is located 1.5 m above the ground, how far from the spout does the water hit the floor?

Ignore all friction and dissipative forces. Round your answer to the nearest 2 decimal places.

Answer 1

(a)By using Bernoull's equation for ideal flow of fluid

Consider 2 points P & Q.

Point P is just below the piston and another point Q is just outside the spout.

So pressure at point P and Q will be$P_a +\frac{mg}{A}$ and $P_a$  respectively where m=20kg is the mass of the piston and A is the area of cross section of piston and $P_a$ is atmospheric pressure .

Now apply Bernoulli's Equation between P and Q.

$P_p+\frac{1}{2}\rho V_p^2 + \rho g h_p = P_Q + \frac{1}{2}\rho V_Q^2 + \rho g h_Q$

From equation of continuity

Area of cross section of piston and Point P is very large compare to at that of Q so we can assume $V_p$ =0.

$h_p$  and $h_Q$  are 0.5m and 0.15m from the base of container respectively.

Now

$P_a + \frac{mg}{A} + \rho g (0.5m) = P_a + \frac{1}{2} \rho V_Q^2 +\rho g (.015m)$

after solving this equation

$V_Q = \lbrack \frac{2mg}{\rho A}+ 2g(0.5-0.15)\rbrack^\frac{1}{2}$

$V_Q= 3.28\frac{m}{sec}$

(b) when height of spout is 1.5m from the ground

then initial distance where water hit the floor is given by projectile motion.

X= horizontal speed of water coming out from spout * time taken to hit the ground

time taken will be $t=(\frac{2h}{g})^\frac{1}{2}$ where h= 1.5m so t= 0.553 sec

now distance will be equal to $\frac{3.28\frac{m}{sec}}{0.553sec}$ = 1.81m

water(initial) coming out from spout will hit the surface at distance 1.81m from wall of the container.