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.

Explanation

Let the total distance be d.

For the first half of the distance, `A` to `B`:
`t_(0)=(d//2)/v_(0)=d/(2v_(0))`
For the next half of the distance, `B` to `C`, let the time be t.
`d_(1)=v_(1)t/2, d_(2)=v_(2)t/2`
`d_(1)+d_(2)=d/2 implies ((v_(1)+v_(2))t)/2=d/2`
`implies t=d/(v_(1)+v_(2))`
Total time of journey `=t_(0)+t=d/(2v_(0))+d/(v_(1)+v_(2))`
Average speed`=d/(d/(2v_(0))+d/(v_(1)+v_(2)))=(2v_(0)(v_(1)+v_(2)))/(2v_(0)+v_(1)+v_(2))`