Question

Two hail stones iron the form of a sphere with radii in the ratio 1:2, fall from a great height through the atmosphere. What will be the ratio of their momenta when they attain their respective terminal velocities?

• A. 1:16
• B. 1;32
• C. 1:1
• D. 1;4

Answer 1

The correct option is B 1;32

The terminal velocity is attained when gravitational force is equal and opposite is given by

$v=\frac{2r^2(\rho -\sigma )g}{9\eta }$. . . . . . . (1)

$v\propto r^2$.

$\frac{v_1}{v_2}=\frac{r_1^2}{r_2^2}$

$\frac{v_1}{v_2}=\frac{1^2}{2^2}=\frac{1}{4}$

Mass, $m=\rho \frac{4}{3}\pi r^3$. . . . . .(2)

$m\propto r^3$

$\frac{m_1}{m_2}=(\frac{r_1}{r_2}{)}^3$

$\frac{m_1}{m_2}=(\frac{1}{2}{)}^3=\frac{1}{8}$

ratio of momentum,

$\frac{P_1}{P_2}=\frac{m_1}{m_2}.\frac{v_1}{v_2}$

$\frac{P_1}{P_2}=\frac{1}{8}.\frac{1}{4}=\frac{1}{32}$

$P_1:P_2=1:32$

The correct option is B.