Question

A large tank filled with water to a height h is to be emptied through a small hole at the bottom.The ratio of times taken for the level of water to fall from $h$ to $h/2$ and from $h/2$ to $0$ is

• A. $\sqrt{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\sqrt{2}-1$
• D. $\frac{1}{\sqrt{2}-1}$

Answer 1

The correct option is C $\sqrt{2}-1$

Consider a cylinder filled with water till height $h$,

By Torricelli's theorem, velocity of ejection is related to height as

$\frac{1}{2}3v^2=3gh$

$\Rightarrow v=\sqrt{2gh}$

$a=\frac{\delta v}{\delta t}=\frac{\delta v}{\delta x}.\frac{\delta x}{\delta t}=\sqrt{2gx}\times \frac{\sqrt{2g}}{2\sqrt{x}}=g$

we have $v=u+at\Rightarrow t=\frac{v-u}{a}$

$\therefore$ ratio of times is $\frac{v_1-u_1}{u_1-u_2}=\frac{\sqrt{2gh}-\sqrt{2g\frac{h}{2}}}{\sqrt{2g\frac{h}{2}}-0}$

$=\sqrt{2}-1$

$v_1=$ velocity at height $h$

$u_1=$ velocity at height $\frac{h}{2}$

$u_2=$ velocity at height $0$