Question

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 figure. 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 (a) the common velocity of $A$ and $B$ at time $t_{0}$ and (b) the spring constant.

Answer 1

(a) Collision between $A$ and $C$ is elastic and mass of both the blocks is same. Therefore, they will exchange their velocities i.e., $C$ will come to rest and $A$ will be moving will velocity $v_{0}$. Let $v$ be the common velocity of $A$ and $B$, then from conservation of linear momentum, we have

$m_{A} v_{0}=\left(m_{A}+m_{B}\right) v$
or $m v_{0}=(m+2 m) v$
or $v=\frac{v_{0}}{3}$
(b) From conservation of energy, we have
$\frac{1}{2} m_{A} v_{0}^{2}=\frac{1}{2}\left(m_{A}+m_{B}\right) v^{2}+\frac{1}{2} k x_{0}^{2}$
or $ \frac{1}{2} m v_{0}^{2}=\frac{1}{2}(3 m)\left(\frac{v_{0}}{3}\right)^{2}+\frac{1}{2} k x_{0}^{2}$
or $ \frac{1}{2} k x_{0}^{2}=\frac{1}{3} m v_{0}^{2} $ or $k=\frac{2 m v_{0}^{2}}{3 x_{0}^{2}}$