Two spherical rain drops with radii in the ratio 1:2 fall from a great height through the atmosphere

Question

Two spherical rain drops with radii in the ratio 1:2 fall from a great height through the atmosphere. The ratio of their momenta after they have attained terminal velocity is\

Professor PC Thomas Classes · Answer

Terminal velocity, $\mathrm{v}_{\mathrm{T}}=\frac{2}{9} \mathrm{r}^{2} \frac{(ho-\sigma) \mathrm{g}}{\eta}$ or $\mathrm{v}_{\mathrm{T}} \propto(\mathrm{r})^{2}$\ $\Rightarrow \frac{\mathrm{v}_{\mathrm{T}_{1}}}{\mathrm{v}_{\mathrm{T}_{2}}}=\left(\frac{1}{2}ight)^{2}$\ $\because$ Density of rain drops are same\ $	herefore \mathrm{m} \propto(\mathrm{r})^{3}$\ $\Rightarrow \frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}}=\left(\frac{1}{2}ight)^{3}$ Therefore ratio of momenta, $\frac{p_{1}}{p_{2}}=\left(\frac{m_{1}}{m_{2}}ight)\left(\frac{v_{1}}{v_{2}}ight)$\ $=\left(\frac{1}{2}ight)^{3}\left(\frac{1}{2}ight)^{2}=\frac{1}{8} 	imes \frac{1}{4}=\frac{1}{32}$\