A block of mass `m` moving with a velocity `v_(0)` collides with a stationary block of mass `M` to which a spring of stiffness `k` is attached, as shown in Fig. Choose the correct alternative(s)

Options

A. The velocity of the centre of mass is `v_(0)`
B. The initial kinetic energy of the system in the centre of mass frame is `1/4((mM)/(M+m))v_(0)^(2)`
C. The maximum compression in the spring is `v_(0)sqrt(((mM)/(m+M) 1/k))`
D. When the spring is in the state of maximum compression, the kinetic energy in the centre of mass frame is zero

Answer

C, D

Explanation

using conservation of linear momentum,
`mv_(0)=(M+m)v`
`v=(mv_(0))/(M+m)` `KE` in centre of mass frame:
`KE_(c)=1/2m(v_(0)-v)^(2)+1/2Mv^(2)=1/2((Mm)/(M+m))v_(0)^(2)`
From energy conservation
`1/2mv_(0)^(2)-1/2(m+M)v^(2)=1/2Kx^(2)`
`implies x=v_(0)sqrt((mM)/(k(M+m)))`