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 the motion.

Options

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

Answer

C

Explanation

`"Average velocity"v_(a)=("Total distance")/("Total time")`
For first half of journey.
Time `t_(1)=("Distance"(x//2))/(v_(0))`
or `t_(1)=(x)/(2v_(0))`
Remaining distance `=x//2`
Mean velocity for this distance `=(v_(1)+v_(2))/(2)`
`therefore` Time for this journey `=(x//2)/((v_(1)+v_(2))//2)=(x)/(v_(1)+v_(2))`
`therefore t_(z)=(x)/((v_(1)+v_(2))`
for the whole journey,
Distance=x
Time `=t_(1)+t_(2)=x(v_(1)+v_(2)+2v_(0))/(2v_(0)(v_(1)+v_(2))`
`therefore v_(a)=(x xx2v_(0)(v_(1)+v_(2)))/(x(v_(1)+v_(2)+2v_(0)))=(2v_(0)(v_(1)+v_(2)))/((v_(1)+v_(2)+2v_(0)))`