Question

Two uniform spherical balls of the same density and radii $r$ and $2r$ are dropped in air and are falling vertically downward. If the terminal velocity of the ball of radius $r$ is $1\text{cm/s}$, then the terminal velocity of the ball of radius $2r$ will be

• A. $1\text{cm/s}$
• B. $4\text{cm/s}$
• C. $3\text{cm/s}$
• D. $2\text{cm/s}$

Answer 1

The correct option is B $4\text{cm/s}$

Given:
Terminal velocity of the ball of radius $r$, $v_{T_1}=1\text{cm/s}$
Terminal velocity of the ball of radius $2r$, $v_{T_2}=?$

Terminal velocity is achieved when net force on the ball is zero.
$\Rightarrow F_{net}=0$
$\Rightarrow mg=F_D$
$(F_D=$ drag force due to air $)$
$\Rightarrow mg=6\pi r\eta v_T$
$\Rightarrow \frac{4}{3}\pi r^3\times \rho \times g=6\pi r\eta v_T$
$\Rightarrow v_T=\frac{2\rho g{r}^2}{9\eta }$
For same material of ball,
$v_T\propto r^2$
So, $\frac{v_{T_1}}{v_{T_2}}={\left(\frac{r_1}{r_2}\right)}^2$
$\Rightarrow \frac{1}{v_{T_2}}={\left(\frac{r}{2r}\right)}^2$
$\Rightarrow v_{T_2}=4\text{cm/s}$