A man traversed half the distance with a velocity `v_(0)`. The remaining part of the distance was covered with velocity `v_(1)`. For half the time and with velocity `v_(2)` for the other half of the time . Find the average speed of the man over the whole time of motion.
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Options

A. `(sqrt(63)V_(0))/(3)`
B. `(sqrt(71)V_(0))/(3)`
C. `(sqrt(73)V_(0))/(3)`
D. `(sqrt(87)V_(0))/(3)`

Answer

C

Explanation

Take `x` axis along the incline and `y` direction perpendicular to it.

Velocity of rain relative to the man is perpendicular to the incline in the case (i.e., along the umbrella stick. This keeps canopy perpendicular to the rainfull and provides maximum safety.
`overset(vec)V_(rm) = overset(vec)V_(r) - overset(vec)V_(m)`
`=(V_(x)hatt + V_(y)hat(j))-(V_(0)hati) = (V_(x) - V_(0))hati+V_(y)hatj`
Since `overset(vec)V_(rm)` has no `x` component.
When the man is walking up. `overset(vec)V_(rm)` us directed vertically downward.
`overset(vec)V_(rm) = (V_(x)hat(i) + V_(y)hat(i)) - (V_(0)hat(i))`
`= V_(0)hati + V_(y)hatj + V_(0)hati`
`= 2V_(0)hati + V_(y)hatj`
From diagram

`tantheta = (2V_(0))/(V_(y))`
`(3)/(4)(2V_(0))/(V_(y)) rArr V_(y) = (8V_(0))/(3)`
`:. V_(r) = sqrt(V_(x)^(2) + V_(y)^(2)) = sqrt((73)/(3))V_(0)`