A smooth rope of mas `m` and length `L` lies in a heap on a smooth horizontal floor, with one end attached to a block of mass `M`. The block is given a sudden kick and instantaneously acquires a horizontal velocity of magnitude `V_(0)` as shown in figure 1. As the block moves to right pulling the rope from heap, the rope being smooth, the heap remains at rest. At the instant block is at a distance `x` from point `P` as shown in figure `-2( P` is a point on the rope which has just started to move at the given instant `)`, choose correct options for next three question.

The speed of block of mass `M` is

Options

A. `(m^(3))/(L)(V_(0)^(2))/((M+(m)/(L)x)^(3))`
B. `(mM^(2))/(L)(V_(0)^(2))/((M+(m)/(L)x)^(3))`
C. `(m^(4))/(ML)(V_(0)^(2))/((M+(m)/(L)x)^(3))`
D. `(M^(2))/(L)(V_(0)^(2))/((M+(m)/(L)x)^(3))`

Answer

B

Explanation

The mass of moving material is `M+(m)/(L)x`
From conservation of momentum `MV_(0)=(M+(m)/(L)x)V`
`:. `Velocity of moving block and moving rope is
`V=(MV_(0))/(M+(m)/(L)x)`
`(B)` The acceleration of moving block is
`a=-v(dv)/(dx)=-(MV_(0))/((M+(m)/(L)x)^(2))xx(m)/(L)(dx)/(dt)`
`=-(m)/(L)(M^(2)V_(0)^(2))/((M+(m)/(L)x)^(3))`