A person standing on a road has to hold his umbrella at `60^(0)` with the verticcal to keep the rain away. He throws the umbrella an starts running at `20 ms^(-1)`. He finds that rain drops are hitting his head vertically. Find the speed of the rain drops wigh respect to (a) the road (b) the moving person.

Explanation

`vec v_(r g) = vec v_(r m) + vec v_(m g) rArr vec v_(r m) = vec v_(r g) - vec v_(m g) = vec v_(r g) + (- vec v_(m g))`
When man is not moving he is observing actual velocity of the rain.
`vec v_("rain") = v sin 60^@ hat i - v cos 60^@ hat j`
=`(sqrt(3))/(2) v hat i - (v)/(2) hat j (m s^-1)` ...(i)
When the man starts running with speed `20 m s^-1`, rain appears to fall vertically as seen by man.
Hence, the velocity of rain with respect to man
`vec v_(r,m) = vec v_("rain") - vec v_(man)` ...(ii)
Let the velocity rain with respect to man has magnitude
`|vec v_(r,m)| v' rArr -v' hat j = ((sqrt(3))/(2) v hat i - (v)/(2) hat j) - 20 hat i`
`-v' hat j =(-20 + (sqrt(3))/(2) v) hat i - (v)/(2) hat j` ...(iii)
Comparing left side and right side terms,
`-20 + (sqrt(2))/(2) v = 0 rArr v = (40)/(sqrt(3)) (m s^-1)`
`v' = (v)/(2)= (1)/(2) ((40)/(sqrt(3))) = (20)/(sqrt(3))(m s^-1)`
Hence, the actual velocity of rain (from (i)),
`vec v_(rain) = (sqrt(3))/(2) ((40)/(3)) hat i - (1)/(2) ((40)/(sqrt(3))) hat j (m s^-1)`
=`20 hat i - (20)/(sqrt(3)) hat j (m s^-1)`
Hence, magnitude of actual velocity of rain,
`|vec v_r| = (40)/(sqrt(2))(m s^-1)`
And the magnitude of velocity of rain with respect to man,
`|vec v_(r,m)| = (20)/(sqrt(3)) (m s^-1)`
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.
Method 2 : Given `theta = 60^@` and velocity of person `vec v_P = vec(OA) = 20 m s^-1`.
This velocity is same as the velocity of person w.r.t. ground.
First of all let is see how the diagram works out.
`vec v_(r P) = vec(OB)` = Velocity of rain w.r.t. person.
`vec v_r = vec(OC)` = Velocity of rain w.r.t. earth
Values of `vec v_r and vec v_(r P)` can be obtained by using simple trigonometric relation.
(a) Speed of rain drops w.r.t. earth `= vec v_r = vec(OC)`
From `Delta OAB, (CB)/(OC) = sin 60^@`
`rArr O C = (C B)/(sin 60^@) = (20)/(sqrt(3)//2) = (40)/(sqrt(2)) = (40 sqrt(3))/(3) ms^-1`
(b) Speed of rain w.r.t. the person, `vec v_(r P) = vec(O B)`
From, `(O B)/(C B) = cot 60^@`
`rArr OB = CB cot 60^@ = (20)/(sqrt(3)) = (20 sqrt(3))/(3) m s^-1`.
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