Question

A body of density $\rho$ is dropped from rest from a height h into a lake of density $\sigma (\sigma >\rho )$ The maximum depth the body sinks inside the liquid is (neglect viscous effect of liquid):

• A. $\frac{h\rho }{\sigma -\rho }$
• B. $\frac{h\sigma }{\sigma -\rho }$
• C. $\frac{h\rho }{\sigma }$
• D. $\frac{h\sigma }{\rho }$

Answer 1

The correct option is A $\frac{h\rho }{\sigma -\rho }$

Velocity of the ball as it reaches water surface is $\sqrt{2gh}$
Net upward force acting on the body $F=\sigma Vg-\rho Vg$
$\Rightarrow ma=\sigma Vg-\rho Vg$
$\Rightarrow \left(\rho V\right)a=Vg(\sigma -\rho )$
$\Rightarrow a=g\left(\frac{\sigma -\rho }{\rho }\right)$ in upward direction
lets say it sinks to the depth $H$, at this instant final velocity becomes $v=0$, so we have
$0=(\sqrt{2gh}{)}^2-2\times g\left(\frac{\sigma -\rho }{\rho }\right)\times H$ $\because v^2=u^2+2as$
$\Rightarrow H=\left(\frac{h\rho }{\sigma -\rho }\right)$