Question

A small body of density ${\rho }^{\prime }$ is dropped from rest at a height $h$ into lake of density $\rho$, where $\rho >{\rho }^{\prime }$. Which of the follwing statements is/are neglected (neglected viscosity)?

• A. The speed of the body just entering the lake is $\sqrt{2gh}$.
• B. The body in the lake experience upward acceleration equal to $\left\{\right(\rho /{\rho }^{\prime })-1\}g$.
• C. The maximum depth to which the body sinks in the lake is $h{\rho }^{\prime }/\left({\rho }^{\prime }/{\rho }^{\prime }\right)$.
• D. The body does not come back to the surface of the lake.

Answer 1

Answer: (A), (B), (C)

The correct options are
A The speed of the body just entering the lake is $\sqrt{2gh}$.
B The body in the lake experience upward acceleration equal to $\left\{\right(\rho /{\rho }^{\prime })-1\}g$.
C The maximum depth to which the body sinks in the lake is $h{\rho }^{\prime }/\left({\rho }^{\prime }/{\rho }^{\prime }\right)$.

Velocity of the ball as it reaches the water surface is equal to $\sqrt{2gh}$. Therefore option $\left(A\right)$ is correct.

Upward velocity acting on the body

$F=\rho Vg-{\rho }^{\prime }Vg$

$\Rightarrow ma=\rho v_q-{\rho }^{\prime }vg$

$\Rightarrow \left({\rho }^{\prime }V\right)a=\rho Vg-{\rho }^{\prime }Vg$

$\Rightarrow a=\frac{g(\rho -{\rho }^{\prime })}{{\rho }^{\prime }}$

$\Rightarrow a=\frac{g(\rho -{\rho }^{\prime })}{{\rho }^{\prime }},$ in upward direction

Let the body sink to a depth of H. Final velocity $v=0.$ Using third equation of kinematics,

$v^2=u^2+2as$

$\Rightarrow 0=(\sqrt{2gh}{)}^2+2\frac{g(\rho -{\rho }^{\prime })}{{\rho }^{\prime }}$

$\Rightarrow H=\frac{h{\rho }^{\prime }}{(\rho -{\rho }^{\prime })}$

∴ options (B) and (C) are correct