Two blocks A and B, each of mass m, are connected by a masslesss spring of natural length L and spring constant K. The blocks are initially resting on a smooth horizontal floor with the spring at its natural length, as shown in fig. A third identical block C, also of mass m, moves on the floor with a speed v along the line joining A and B, and collides elastically with A. Then

Options

A. The frequency of oscillation of the system `AB` is `(1)/(2pi)sqrt((2K)/(m))`
B. The `K.E.` of the system at maximum compression of the spring is `mv^(2)//4`
C. The maximum compression of the spring is `v sqrt((m)/(K))`
D. The maximum compression of the spring is `vsqrt((m)/(2K))`

Answer

A, B, D

Explanation

Let `v_(1)` is the velocity acquired by `A`and `B` then
`mv = mv_(1) +mv_(1) i.e., v_(1) = (v)/(2)`
so, `(1)/(2) mv'^(2) =(1)/(2)mv_(1)^(2) +(1)/(2)mv_(1)^(2) +(1)/(2)kx^(2)`
where `x` is displacement i.e., `x=v(M)/(2K)` ltbr. So, choice (d) is correct.
At maximum compression, `KE.` Of `A-B` system is
`=(1)/(2)mv_(1)^(2) +(1)/(2)mv_(1)^(2) =m_(1)^(2) =(1)/(4)mv^(2)`
so, choice (b) is correct.