A body travelling along a straight line traversed one half of the total distance was covered with a velocity v0. The remaining part of the distance was covered with a velocity v1, for half of time and with velocity v2 for the other half of time. Find the mean velocity averaged over the whole time of motion....
Answers
solution : Let length of path is x.
half of path is travelled by velocity Vo.
so, time taken to cover distance, t1 = (x/2)/Vo = x/2Vo......(1)
now, reaming distance = x/2, is covered with velocity v1 for half time and v2 for other half time.
let time time to cover the distance x/2 is t
then, distance covered with velocity v1 =v1 × (t/2)
and distance covered with velocity v2. =v2 × (t/2)
so, average velocity of particle for remaining (x/2) distance , V' = (v1 × t/2 + v2 × t/2)/(t/2 + t/2) = (v1 + v2)/2
now, time taken to cover (x/2) distance with velocity (v1 + v2)/2 , t2 = (x/2)/(v1 + v2)/2 = x/(v1 + v2).....(2)
now average velocity for whole time of motion, V = total distance/total time taken
= x/(t1 + t2)
from equations (1) and (2),
= x/[x/2Vo + x/(v1 + v2)]
= 2Vo(v1 + v2)/(2Vo + v1 + v2)
hence, average velocity is