Relative Motion

RAIN-UMBRELLA PROBLEM

1.Rain is falling vertically downwards with velocity \(\vec U\).A person is walking on a road with velocity  \(\vec V\).In order to protect himself from rain he must hold his umbrella in the direction of relative velocity of rain with respect to him.Magnitude of relative velocity of rain with respect to the person is \( |\overrightarrow U - \overrightarrow V | = \sqrt {U^2 + V^2 } \)  

Angle made by the umbrella to the vertical is ‘\(\theta\)’ then tan\(\theta\)=\( \frac{V} {U} \)  and \(\theta\)=\( \tan ^{ - 1} \left( {\frac{V} {U}} \right) \)
Here sin\(\theta\)=\( \frac{V} {{\sqrt {U^2 + V^2 } }} \)  and cos\(\theta\) =\( \frac{U} {{\sqrt {U^2 + V^2 } }} \)