A point 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 mean velocity of the point averaged over the whole time of motion.

Options

A. `(v_(0) + v_(1) + v_(2))/(3)`
B. `(2v_(0) + v_(1) + v_(2))/(3)`
C. `(v_(0) + 2v_(1) + 2v_(2))/(3)`
D. `(2v_(0) (v_(1) + v_(2)))/((2v_(0) + v_(1) + v_(2)))`

Answer

D

Explanation

Let the total distance be d. Then for first half distance, time `=(d)/(2v_(0))`, next distance `=v_(1)t` and last half distance `=v_(2)t`
`:. v_(1) t+ v_(2)t + (d)/(2), t = (d)/(2(v_(1) + v_(2)))`
Now average speed `t= (d)/((d)/(2v_(0)) + (d)/(2(v_(1) + v_(2))) + (d)/(2(v_(1)+ v_(2))))= (2v_(0) (v_(1) + v_(2)))/((v_(1) + v_(2)) + 2v_(0))`