Question

A point traversed half of the distance with a velocity vo. The remaining part of the distance was covered with velocity v1 for half the time and Wii velocity v2 for the other half of the time. The mean velocity of the point averaged over the whole time of motion is

Answer 1

Let the total distance be D.

Let the point transversed half the distance by velocity V1.

Then the time required by it to do so = (D/2)/V1 = D/2V1

The remaining half was transversed by travelling at V2 for halftime and V3 velocity for another half time.

Let the time to cover the second half of the journey = t

Then V2t/2+V3t/2= D/2(Second Half of journey)

So t= D/(V2+V3)

The total time required for the journey is (D/2V1)+(D/(V2+V3))

The Average or mean Velocity = D/( (D/2V1) + (D/(V2+V3)) ) = 1/( (1/2V1) + (1/(V2+V3)) ) =2V1(V2+V3)/(2V1+V2+V3)----------(ans)

Hope IT Helps

Answer 2

ANSWER:

Let s be the total distance traversed by the point and t1 the time taken to cover half the distance. Further let 2t be the time to cover the rest half of the distance.

Therefore s/2=v0t1 or t1=s/2v0 ...........(1)

and s/2=(v1+v2)t or 2t=s/v1+v2 ........(2)

Hence the sought average velocity

<v>=s/t1+2t

     =s/[s/2v0]+[s/(v1+v2)]

     =2/v0(v1+v2)v1+v2+2v0