Question

A tank is filled upto height $h$ with a liquid and is placed on a platform of height $h$ from the ground. To get maximum range $x_m$, a small hole is punched at a distance $y$ from the free surface of the liquid. Then find the value of maximum range ($x_m$).

• A. $x_m=h$
• B. $x_m=2h$
• C. $x_m=4h$
• D. $x_m=\frac{h}{2}$

Answer 1

The correct option is B $x_m=2h$


Velocity of efflux $V=V_x=\sqrt{2gy}$
Let $t$ be the time taken by fluid to hit the ground.
Using equation of motion
$S_y=u_{y}t+\frac{1}{2}{a}_y{t}^2$
$u_y=0,a_y=-g,S_y=-(2h-y)$
$\Rightarrow t=\sqrt{\frac{2(2h-y)}{g}}\dots \left(1\right)$

Horizontal range $=V_{x}t$
$R=\sqrt{2gy}\sqrt{\frac{2(2h-y)}{g}}\dots \left(2\right)$
For maximum range, $\frac{dR}{dy}=0$
$\Rightarrow \frac{1}{4\sqrt{y(2h-y)}}(8h-8y)=0$ $[y\ne 0,2h$]
$\Rightarrow y=h$
Substituting in (2), $R_{\text{max}}=x_m=\sqrt{4h^2}=2h$