Question

In a cylindrical vessel containing liquid of density $p$, there are two holes in the side wall at heights $h_1$ and $h_2$ respectively such that the range of efflux at the bottom of the vessel is same. The height of a hole, for which the range of efflux would be maximum, will be:

• A. $(h_2-h_1)$
• B. $\frac{h_2-h_1}{2}$
• C. $h_1+h_2$
• D. $\frac{h_1+h_2}{2}$

Answer 1

The correct option is D $\frac{h_1+h_2}{2}$

let height of vessel = H.

from, Bernoulli's theorem.

velocity of liquid at $h_1,V_1=\sqrt{2g(H-h_1)}$

Range $R_1=v_{1}t$

for $t$,

$h_1=\frac{1}{2}\times g\times t^2$

$t=\sqrt{\frac{2h_1}{g}}$

$R_1=V_{1}t=2\sqrt{(H-h)}{h}_1$

For $h_2$ height, Range $R_2=2\sqrt{(H-h_2)h_2}$

$\because R_1=R_2$

$2\sqrt{(H-h_1)h_1}=2\sqrt{(H-h_2)h_2}$

0r

$H(h_1-h_2)=(h_1+h_2)(h_1-h_2)$

Or

$H=h_1+h_2$

now, let $h_1$ height is ${}^{\prime }{y}^{\prime }$ height below than ${}^{\prime }{H}^{\prime }$

$R=2\sqrt{(H-y)y}$

$\frac{d{R}^2}{dy}=\frac{d}{dy}(Hy-y^2)$

$H-2y=0$

or $y=H/2$

for $max$ value $y=\frac{h_1+h_2}{2}$

Correct option is (D).