Question

Water is filled in a uniform container of area of cross section A. A hole of cross section area a (<< A) is made in the container at a height of 20 m above the base. Water streams out and hits a small block placed at some distance from container. With what speed $\left(inm{s}^{-1}\right)$ the block should be moved such that water streams always hits the block. (Given $\frac{a}{A}=\frac{1}{20}$). (Take $g=10m{s}^{-2}$) ___

Answer 1

Velocity of efflux $v=\sqrt{2gy}$
Range $x=\sqrt{2gy}\times \sqrt{\frac{2h}{g}}$
The velocity of the block must be $\left(\frac{dx}{dt}\right)$.
$\therefore V_b=\frac{dx}{dt}=\sqrt{\frac{2h}{g}}\times \sqrt{2g}\times \frac{1}{2\sqrt{y}}\frac{dy}{dt}$
$V_b=\frac{\sqrt{h}}{\sqrt{y}}.\frac{dy}{dt}\dots \dots \left(i\right)$
Using equation of continuity
$\frac{Ady}{dt}=a\sqrt{2gy}\dots \dots \left(ii\right)$
equation (i) and (ii)
$V_b=\sqrt{\frac{h}{y}}\times \frac{a}{A}\sqrt{2gy}$
$V_b=\sqrt{2gh}\times \frac{a}{A}=20\times \frac{1}{20}=1m{s}^{-1}$