Question

Water is filled in a uniform container of area of cross section $A$. A hole of cross section area $a\left(«A\right)$ is made in the container at a height of $20m$ above the base. Water streams out and hits a small block on surface at some distance from the container. The block is moved on the surface in such a way that the stream always hits the block. The initial velocity of the block (in $m/s$) is (Given $\frac{a}{A}=\frac{1}{20}$; Take $g=10m{s}^{-2}$) (answer upto two decimal places)

Answer 1

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