Why does a person hold his umbrella at some angle with the vertical while he is walking on road? Assume that rain is falling vertically downwards.

This happens due to the relative velocity. The relative velocity of rain with respect to the person is the difference in the velocities of the rain and the person. \[\text{i.e.}\;\; v_{\text{rel}} = v_{\text{rain}}-v_{\text{per}}\] [From Triangle Law of Vector Addition]

Why does a person hold his umbrella at some angle with the vertical while he is walking on road? Assume that rain is falling vertically downwards.

It means that the magnitude as well as direction of the velocity of rain as observed by that person is different. This gives the displacement also different from the original, both in terms of magnitude and direction. So, the person sees rain drops coming in direction AC rather than AB.

Let $R$ be the resultant which makes an angle $\theta$ with the vertical. Then, \[R=v_{\text{rain}}^2+v_{\text{per}}^2\] and, \[\tan\theta=\frac{v_{\text{per}}}{v_{\text{rain}}}\] \[\theta = \tan^{-1}\left(\frac{v_{\text{per}}}{v_{\text{rain}}}\right)\]

Hence, rain strikes the person by making an angle $\theta$ with the vertical. So, the person holds his umbrella towards A, i.e. in inclined manner.

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