Question

In a cylindrical water tank there are two small holes $Q$ and $P$ on the wall at a depth of $h_{1}$ from the upper level of water and at a height of $h_{2}$ from the lower end of the tank, respectively, as shown in the figure. Water coming out from both the holes strike the ground at the same point. The ratio of $h_{1}$ and $h_{2}$ is

• A. $1$
• B. $2$
• C. $ > 1$
• D. $ < 1$

Answer 1

Answer: (A)

The two streams strike at the same point on the ground.

$R_{1}=R_{2}=R$
$u_{1} t_{1}=u_{2} t_{2}$ ...(i)
where $u_{1}=$ velocity of efflux at $Q=\sqrt{\left(2 g h_{1}\right)}$
and $u_{2}=$ velocity of efflux at $P=\sqrt{\left[2 g\left(H-h_{2}\right)\right]}$
$t_{1}=$ time of fall of water stream through $Q$ is
$=\sqrt{\frac{2\left(H-h_{1}\right)}{g}}$
$t_{2}=$ time of fall of the water stream through $P=\sqrt{\frac{2 h_{2}}{g}}$
Putting these values is Eq. (i), we get
$\left(H-h_{1}\right) h_{1}=\left(H-h_{2}\right) h_{2}$
or $\left[H-\left(h_{1}+h_{2}\right)\right]\left[h_{1}-h_{2}\right]=0$
$H=h_{1}+h_{2}$ is irrelevant because the holes are at two different heights.
Therefore, $h_{1}=h_{2}$ or $h_{1} / h_{2}=1$