Question

A leakage begins in water tank at position $\text{P}$ as shown in the figure. The initial gauge pressure (pressure above that of the atmosphere) at $\text{P}$ was $5\times {10}^5{\text{N/m}}^2$. If the density of water is $1000{\text{kg/m}}^3$, the initial velocity with which water gushes out is approximately.

• A. $3.2\text{m/s}$
• B. $32\text{m/s}$
• C. $28\text{m/s}$
• D. $2.8\text{m/s}$

Answer 1

The correct option is B $32\text{m/s}$

Given:
density of water, $\rho =100{\text{kg/m}}^3$

Gauge pressure, $\Delta p=5\times {10}^5{\text{N/m}}^2$


$p_1=p_{\text{atm}}+\Delta p;p_2=p_{\text{atm}}$

$v_1=0;v_2=V$
[ initial velocity means the water just start]

On applying Bernoulli's equation at $\text{P}$ [inside and outside the hole]

$p_1+\rho g{h}_1+\frac{1}{2}\rho v_1^2=p_2+\rho g{h}_2+\frac{1}{2}\rho v_2^2$

putting the value from above data, we get

$\Rightarrow \Delta p=\frac{1}{2}\rho V^2$

$\Rightarrow 5\times {10}^5=\frac{1}{2}\times 1000\times V^2$

$\Rightarrow V=10\sqrt{10}=31.6\text{m/s}\approx 32\text{m/s}$

Hence, option $\left(b\right)$ is correct.

$\begin{array}{c}\text{Why this question ?}\\ \text{Concept - As water starts to gush out, initial velocity}\\ \text{inside the water just after the leakage will be zero.}\\ \text{After some time, velocities of water inside and outside}\\ \text{will be equal which will lead to equal pressure inside}\\ \text{and outside (equal to atmospheric pressure).}\end{array}$