Question

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 figure. Choose the correct alternative(s).

• A. The velocity of centre of mass is $v_0$.
• B. The initial kinetic energy of the system in the centre of mass frame is $\frac{1}{4}\left(\frac{mM}{m+M}\right)v_0^2$ .
• C. The maximum compression in the spring is $v_0\sqrt{\frac{mM}{m+M}\frac{1}{k}}$ .
• D. When the spring is in the state of maximum compression, the kinetic energy in the centre of mass frame is zero.

Answer 1

Answer: (C), (D)

When the spring is in the state of maximum compression, the kinetic energy in the centre of mass frame is zero.

Velocity of centre of mass will be,
$\begin{array}{}{V}_{com}=\frac{m{V}_o+M\left(0\right)}{m+M}=\frac{m{V}_o}{m+M}& \text{(1)}\end{array}$
From COM frame of reference,
Initial kinetic energy $\left(K{E}_i\right)$ of the system,
$K{E}_i=\frac{1}{2}m(V_o-V_{com}{)}^2+\frac{1}{2}M{V}_{com}^2$
Using the value of $V_{com}$ from $\left(1\right)$, we get
$\begin{array}{}K{E}_i=\frac{mM{V}_o^2}{2(m+M)}& \text{(2)}\end{array}$
Hence option 2 is incorrect.
Maximum compression in the spring will occur when all kinetic energy of the system converts into potential energy of the spring.
Initial Kinetic energy = Potential energy of the spring
$\frac{mM{V}_o^2}{2(m+M)}=\frac{1}{2}k{x}^2$
$x=v_0\sqrt{\frac{mM}{k(m+M)}}$
Hence option 3 and 4 is correct.