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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Explanation

Let s = total distance traversed by the particle, and `t_(1)` = time taken to travel half the distance.
? `(s)/(2) =v_(0)n t_(1)," or "t_(1) =(s)/(2v_(0))`
Now, let 2t be the time taken to cover the remaining distance `(s)/(2)`. Let `s_(1)` distance be covered with velocity `v_(1)`.
? `s_(1) =v_(1) t`
Similarly, t be the time taken by the particle to travel `s_(2)` distance with velocity `v_(2)`
`s_(1) + s_(2) =(v_(1)+v_(2)) t" but "s_(1//2) =s//2 rarr (s)/(2) = (v_(1)+v_(2)) t rarr 2t =(s)/(v_(1)+v_(2))`
Now average speed `=("total distance")/("total time") =(s)/(t_(1)+2t) =(s)/((s)/(2v_(0))+(s)/(v_(1)+v_(2)))=(2v_(0)(v_(1)+v_(2)))/(v_(1)+v_(2)+2v_(0))`