Question

A spherical ball of radius $1\times {10}^{-4}m$ and density ${10}^{4}kg/m^3$ falls freely under gravity through a distance $h$ before entering a tank of water. If after entering the water the velocity of the ball does not change, find $h$ in m (Write answer to nearest interger). (The viscosity of water is $9.8\times {10}^{-6}{N-s/m}^2$)

Answer 1

If the ball's velocity didn't change after entering into water. that means ball achieved terminal velocity as it touched the water surface.
And the terminal velocity $V_c$ of the body of radius $r$, density
$\rho$ falling through a medium of density ${\rho }_o$ is given by
$V_c=\frac{2r^2(\rho -{\rho }_o)g}{9\eta }$ where $\eta$ is the coefficient of viscosity of medium.
$V_c=\frac{2\times ({10}^{-4}{)}^2\times ({10}^4-{10}^3)\times 9.8}{9\times 9.8\times {10}^{-6}}=20m/s$
Velocity attained by the ball after descending $h$ height from rest is given by
$v^2=2gh\because v^2=u^2=2aS$
$\Rightarrow h=\frac{v^2}{2g}$
$\Rightarrow h=\frac{{20}^2}{2\times 9.8}=20.4m$