A particle travels along a straight line. It covers halg the distance with a speed (v). The remaining part of the distance was covere with speed `v_(1)` for half the time and with speed `v_(2)` for the other half the time . Find the average speed of the particle over the entire motion.

Explanation

Let (S) be the total distance to betravelled by the particle.
Time taken by particle to travel halgf the distance
`(S//2)` is `t_(1) =(S//2)/v =S/(2v)`
Let `2t_(2)` be the time takenby particle to cover the remainig distance `S//2`
Distance travelled by particle in time `t_(2)` while travelling with speed `v_(1)` is
Distance travelled by particle in time `t_(2)` while travelling with speed `v_(2` is
`S_(2) =v_(1) t_(2)`
`S_(1) +S_(2) =v_(1) t_(2) + v_(2) t_(2)
` But `S_(1) +S_(2) =S/2 =v_(1) t_(2) +v_(2) t_(2) + (v_(1) +v_(2)) t_(2)`
or `2t_(2) =S/((v_(1) +v_(2))`
Average speed `=(total distance travelled)/(total time talke)`
`=S/(t_(1) +2t_(2)) =S/((S//2 v) + S//(v_(1) +v_(2))`
`=(2 v(v_(1) + v_(2))/(v_(1) +v_(2) +2v)`.