Question

Two blocks A and B of masses m and $2m$, respectively, are connected with the help of a spring having spring constant, k as shown in Fig. Initially, both the blocks are moving with same velocity v on a smooth horizontal plane with the spring in its natural length. During their course of motion, block B makes an inelastic collision with block C of mass m which is initially at rest. The coefficient of restitution for the collision is $1/2$. The maximum compression in the spring is?

• A. $\sqrt{\frac{2m}{k}}$
• B. Will never be attained
• C. $\sqrt{\frac{m}{12k}}v$
• D. $\sqrt{\frac{m}{6k}}v$

Answer 1

The correct option is D $\sqrt{\frac{m}{6k}}v$

For collision of B and C
Solving above equations, $v_1=\frac{v}{2}$ and $v_2=v$
Now for blocks A and B plus spring system, using reduced mass concept, and applying work-energy theorem, let maximum compression in spring be $x_0$ and at the time of maximum compression relative velocity of blocks be zero. Reduced mass is gien by
$\mu =\frac{m\times 2m}{3m}=\frac{2m}{3}$
$0-\frac{\mu \times (v-v/2{)}^2}{2}=-\frac{k{x}_0^2}{2}$
$\Rightarrow x_0=\left(\sqrt{\frac{m}{6k}}\right)v$.