Question

Water is filled in a container up to height $100\,m.$ A small hole is made at height $80\,m$ from the base of the container such that water start flowing from the container. The distance of the point where the water strikes the ground from the base of the container is measured. Now an external force is applied on the piston and the process is repeated with the initial level of the water at $100m.$ The external force applied on the piston to increase the distance of water from the base of the container to $3$ times the earlier measured value is $\alpha kN.$ The value of $\alpha $ is

Answer 1

From Torricelli's theorem,
$v=\sqrt{2 g \times \left(20\right)}$
$R=\sqrt{2 g \times \left(20\right)}\times \sqrt{\frac{2 \times 80}{ g}}$
$=2\times 40=80m$
Now,
$R^{'}=3\times 80=240m=v^{'}\sqrt{\frac{2 \times 80}{10}}$
$240=v'\left(4\right)$
$v'=60ms^{- 1}$
$\left(P_{0} + \frac{F}{A}\right)+\rho gh=P_{0}+\frac{1}{2}\rho v'^{2}\Rightarrow \frac{F}{A}=\frac{1}{2}\rho v'^{2}-\rho gh\Rightarrow \frac{F}{A}=\frac{1}{2}\times 1000\times \left(60\right)^{2}-1000\times 10\times 20\Rightarrow F=0.01\times 1000\left(1800 - 200\right)=10\times 1600=16kN$