Two bodies A and B of masses m and 2 m respectively are placed on a smooth floor. They are connected by a spring. A third body C of mass m moves with velocity `v_0` along the line joining A and B and collides elastically with A as shown in Fig.

At a certain instant of time `t_0` after collision, it is found that the instantaneous velocities of A and B are the same. Further at this instant the compression of the spring is found to be `x_0`. Determine (i) the common velocity of A and B at time `t_0`, and (ii) the spring constant.

Answer

(i) `(v_(0))/(3)` (ii) `(2mv_(0)^(2))/(3x_(0)^(2))`

Explanation

After collision `U_(A) = V_(0)`
(i) When `(v_("inst")_(A) = (v_("inst")_(B) rArr mv_(0) = 3mv rArr v = v_(0)//3`
(ii) COME
`rArr (1)/(2) mv_(0)^(2) = (1)/(2)(3m)v^(2) + (1)/(2) kx_(0)^(2)`
`k = (2m)/(3) ((v_(0))/(x_(0)))^(2)`