Question

A person standing on a road has to hold his umbrella at ${60}^o$ with the vertical to keep the rain away. He throws the umbrella and starts running at $20m{s}^{-1}$. He find that rain drops are hitting his head vertically. Find the speed of the rain drops with respect to (a) the road and (b) the moving person.

Answer 1

${\overrightarrow{v}}_{rg}={\overrightarrow{v}}_m+{\overrightarrow{v}}_{mg}\Rightarrow {\overrightarrow{v}}_{rm}={\overrightarrow{v}}_{rg}-{\overrightarrow{v}}_{mg}={\overrightarrow{v}}_{rg}+(-{\overrightarrow{v}}_{mg})$

When man is not moving he is observing actual velocity of the rain.

${\overrightarrow{v}}_{rain}=v\sin {60}^o\hat{i}-v\cos {60}^o\hat{j}$

$=\frac{\sqrt{3}}{2}v\hat{i}-\frac{v}{2}\hat{j}\left(m{s}^{-1}\right)$ ...(i)

When the man starts running with speed $20m{s}^{-1}$, rain appears to fall vertically as seen by man.

Hence, the velocity of rain with respect to man.

$\overrightarrow{v_{r.m}}=\overrightarrow{v_{rain}}-\overrightarrow{v_{man}}$ ...(ii)

Let the velocity of rain with respect to man has magnitude

$|{\overrightarrow{v}}_{r.m}|=v^{\prime }\Rightarrow -v^{\prime }\hat{j}=(\frac{\sqrt{3}}{2}v\hat{i}-\frac{v}{2}\hat{j})-20\hat{i}$

$-v^{\prime }\hat{j}=(-20+\frac{\sqrt{3}}{2}v)\hat{i}-\frac{v}{2}\hat{j}$ ...(iii)

Comparing left side and right side terms,

$-20+\frac{\sqrt{3}}{2}v=0\Rightarrow v=\frac{40}{\sqrt{3}}\left(m{s}^{-1}\right)$

$v^{\prime }=\frac{v}{2}=\frac{1}{2}\left(\frac{40}{\sqrt{3}}\right)=\frac{20}{\sqrt{3}}\left(m{s}^{-1}\right)$

Hence, the actual velocity of rain (from (i)),

${\overrightarrow{v}}_{rain}=\frac{\sqrt{3}}{2}\left(\frac{40}{\sqrt{3}}\right)\hat{i}-\frac{1}{2}\left(\frac{40}{\sqrt{3}}\right)\hat{j}\left(m{s}^{-1}\right)$

$=20\hat{i}-\frac{20}{\sqrt{3}}\hat{j}\left(m{s}^{-1}\right)$

Hence, magnitude of actual velocity of rain,

$|{\overrightarrow{v}}_r|=\frac{40}{\sqrt{2}}\left(m{s}^{-1}\right)$

And the magnitude of velocity of rain with respect to man,

$|{\overrightarrow{v}}_{r.m}|=\frac{20}{\sqrt{3}}\left(m{s}^{-1}\right)$

In relative velocity, the observer observes the velocity of an object considering himself at rest.

Consider the example of a man sitting in a moving train and observes the objects outside situated on the ground.