A person walking in a straight line, covers half of the distance to be travelled with a speed of `v_(0).` For half of the time, required to complete the remaining distance, he walks with a speed of `v_(1)` and for the remaining half time his speed is `v_(2)` What is the person's average speed during this complete walk ?

Explanation

Suppose, the total walking distance is d, time taken to travel half of the distance `((d)/(2))` is `t _(1)` and time taken to travel remaining half distance is 2t. Now, distance travelled = average speed `xx` time
`therefore d/2 = v _(0) t _(1)` and hence `t _(1) = (d)/(2v _(0)) " "...(1)`
and `d/2 = v _(1) t + v _(2)t ` (for remaining half distance)
`= (v_(1) +v_(2))t `
`therefore 2t = (d)/(v _(1) + v _(2))`
`(because t = 2t` for that half distance) ...(ii)
Thus, tota time taken to walk total distance d is `t +2t.`
Now the person.s average speed
`lt v gt = (d)/(t _(1) + 2t)`
`= (d)/(((d)/(2 v _(0))) +((d)/( v _(1) + v _(2))))` (From eq. (i) & (ii))
`therefore lt v gt = (2 v _(0) (v _(1) + v _(2)))/( v _(1) + v _(2) + 2v _(0))`