Consider a particle moving along x-axis. Its distance from origin O is described by the co-ordinate x which varies with time. At a time `t_(1)`, the particle is at point P, where its co-ordinate is `x_(1)` and at time `t_(2)`, the particle is at point Q, where its co-ordinate is `x_(2)`. The displacement during the time interval from `t_(1)` to `t_(2)` is the vector from P to Q, the x-component of this vector is `(x_(2) - x_(1))` and all other components are zero.
It is convenient to represent the quantity `x_(2) - x_(1)` the change in x by means of a notation `Delta`, thus `Delta x = x_(2) - x_(1)` and `Delta t = t_(2) - t_(1)`.

The average velocity `bar(V) = (x_(2) - x_(1))/(t_(2) - t_(1)) = (Delta x)/(Delta t)`
A particle moves half the time of its journey with `u`. The rest of the half time it moves with two velocities `v_(1)` and `v_(2)` such that half the distance it covers with `v_(1)` and the other half with `v_(2)`. The net average velocity is: (Assume straight line motion)

Options

A. `(u (v_(1) + v_(2)) + 2 v_(1) v_(2))/(2 (v_(1) + v_(2)))`
B. `(2u(v_(1) + v_(2)))/(2u + v_(1) + v_(2))`
C. `(u (v_(1) + v_(2)))/(2v_(1))`
D. `(2v_(1) v_(2))/(u + v_(1) + v_(2))`

Answer

A

Explanation

Let total time is T
`s_(1) = u xx (T)/(2)`
`(s_(2))/(2) = v_(1) xx t_(1)`
and `(s_(2))/(2) = v_(2) xx t_(2)`
where `t_(1) + t_(2) = (T)/(2)`
Av. Velocity `= ("Total distance")/("Total time")`
`= ((s_(1) + s_(2))/(T))`