A small block of mass m is plced over a long plank of mass M the coefficient of friction between then is `mu` .Ground is smooth At `t = 0` m is given a velocity `v_(1)` and `M` a velocity `v_(2) (gtv_(1))` as shown in Fig. After this `M` is maintained at constant celeration `a(ltmu g)`

Initially there will be some relative motion between the block and the plank ,but after some time relative motion will cease and velocities of both will become
Find the time `t_(0)` when the velocities of both block and plank become same.

Options

A. `(v_(2) - v_(1))/(mu g +a)`
B. `(v_(2) + v_(1))/(mu g -a)`
C. `(v_(2) - v_(1))/(mu g -a)`
D. `(v_(2) + v_(1))/(mu g +a)`

Answer

c

Explanation

For `t lt t_(0): f = mu mg, a_(1) = (f)/(m)= mu g`
Velocity of block at any time
`v_(b)= v_(1) + a_(1)t`
`v_(b)= v_(1) + mu gt`
Velocity of plank at any time `v_(p)= v_(2) +dt`
At `t = t_(0)` both velocity are same
`v_(1) + mu gt_(0) = v_(2) + at_(0) rArr t_(0) = (v_(1) - v_(1))/(mu g - a)`