Question

a particle traversed along a straight line for first half time with velocity V 0​

Answer 1

your complete question is ----> A body travelling along a straight line traversed one half of the total distance with a velocity Vo. The remaining part of the distance was covered with a velocity V1, for half the time and with velocity V2 for the other half of time.Find the mean velocity averaged over the whole time of motion.

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 $\frac{2v_0(v_1+v_2)}{(2v_0+v_1+v_2)}$