Question

rain appears to fall vertically downward to a man running at the rate of 10m/s towards east. if he doubles his speed in the same direction then the rain appears to make an angle of 30 degree with the vertical. find the actual speed and direction of rain wrt ground?

Answer 1

Let the rain be actually falling at a speed v and at an angle alpha w.r.t. ground.
Then, its horizontal component will be v cos(alpha) and vertical component = v sin(alpha)
When the man is running at 10m/s, the rain appears to be falling vertical, i.e. he must have canceled out the horizontal component
10 = v cos(alpha) ...(1)
Now, when he is running at 20m/s ,the rain makes an angle 30 degrees with the vertical, or 60 degrees with the horizontal.
Hence, tan60 = y component/x component = v sin(alpha)/(v cos(alpha) - 20) ..(2)
2 eqns, 2 variables, just solve for alpha and v.