Two hail stones with radii in the ratio of 1: 2 fall from a great height through the atmosphere. Then the ratio of their momenta after they have attained terminal velocity is :

Question

Two hail stones with radii in the ratio of 1: 2 fall from a great height through the atmosphere. Then the ratio of their momenta after they have attained terminal velocity is : (A) 1: 1 (B) 1: 4 (C) 1: 16 (D) 1: 32

Tardigrade Admin · Accepted Answer

Terminal velocity is attained when gravitational force is equal and opposite to atmospheric drag.
It is given by

v = \frac{2}{9} \frac{r ^{2} ( ρ - σ )}{η} g

where

η

is coefficient of viscosity,

r

the radius,

ρ

and

σ

the density.

∴ \frac{v _{1}}{v _{2}} = \frac{r _{1}^{2}}{r _{2}^{2}} = (\frac{1}{2})^{2} = \frac{1}{4}

Ratio of masses of the two hail stones is

\frac{m _{1}}{m _{2}} = \frac{\frac{4}{3} π r _{1}^{3} \cdot ρ}{\frac{4}{3} π r _{2}^{3} \cdot ρ} = \frac{1}{8}

The ratio of momenta

= \frac{p _{1}}{p _{2}} = \frac{m _{1} v _{1}}{m _{2} v _{2}}

= \frac{1}{8} \times \frac{1}{4} = \frac{1}{32}

\Rightarrow

\frac{d ^{2} y}{d t ^{2}} + \frac{k}{m} y = 0

Q. Two hail stones with radii in the ratio of $1 : 2$ fall from a great height through the atmosphere. Then the ratio of their momenta after they have attained terminal velocity is :

1 : 1

1 : 4

1 : 16

1 : 32

Q. Two hail stones with radii in the ratio of 1:2 fall from a great height through the atmosphere. Then the ratio of their momenta after they have attained terminal velocity is :

1 : 1

1 : 4

1 : 16

1 : 32

Q. Two hail stones with radii in the ratio of $1 : 2$ fall from a great height through the atmosphere. Then the ratio of their momenta after they have attained terminal velocity is :