Respected Sir/Madam,please help me out.

A man walking down an incline at the rate of 5 m/s as shown in the figure sees the rain falling directly down on his head.When he walks at the speed of 10 m/s he sees the rain falling at an angle of 30 degrees with the vertical.Find the speed of the rain.

(Figure of Q.19 in this page)

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Please refer to the attached image.

Let M be the point in the inclined plane where from the man is seeing the rain, while moving downwards. Further assume that the velocity of rain is v. In order to get the relative velocity of the raindrops and their apparent direction of falling, laws of vector addition have to be resorted to.

When the man walks at the speed of 5 m/s he sees the rain falling directly down on his head, i.e. along SM, which is actually at an angle 30° with the vertical.

Thus, applying the laws of vector addition,

`tan30° = (5*sin60°)/(v-5*cos60°)`

`rArr 1/sqrt3 = (5*sqrt3/2)/(v-5*1/2)`

`rArr 2v-5=15`

`rArr v =10` m/s

Later, when he walks at the speed of 10 m/s he sees the rain falling on his head at an angle 30°, i.e. along `R_2M` , which is actually at an angle 60° with the vertical.

Thus, according to the laws of vector addition,

`tan60° = (10*sin60°)/(v-10*cos60°)`

`rArr sqrt3 = (10*sqrt3/2)/(v-10*1/2)`

`rArr v-5=5`

`rArr v =10 ` m/s

**Therefore, the raindrops were falling at a speed of 10 m/s.**

**Sources:**

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