Two boxes A and B of masses m and M interconnected by an ideal rope and ideal pulleys are held at rest as shown. When it is released, box B accelerates downwards. Find velocities of box A and B as function of time t after system has been released.

Explanation

We first explore relation between velocities `v_(A)` and `v_(B)` of the boxes A and B, which can be written either by using constrained relation or method of virtual work or by inspection.
`v_(A)= 2v_(B)` ....(i)

Applying impulse momentum principle to box A
`P_(2y) = P_(1y)+sum I_(mp,y) rarr" " mv_B=0+"Mgt"-2Tt` ...(ii)

Applying impulse momentum principle to box B
`P_(2y) = p_(1y) +sumI_(mp,y) rarr" " mv_(B) = 0+Mg t- 2Tt` ...(iii)

From equations (i), (ii) and(iii), we have
` v_(A) = 2((M-2m)/(M+4m))g t` and `v_(B) = ((M-2m)/(M+4m))g t`