Question

A block $C$ of mass m is moving with velocity $v_o$ and collides elastically with block $A$ of mass m and connected to another block $B$ of mass $2m$ through spring of spring constant $k$. What is $k$, if $x_o$ is compression of spring when velocity of $A$ and $B$ is same?

• A. $\frac{m{v}_0^2}{x_0^2}$
• B. $\frac{m{v}_0^2}{2x_0^2}$
• C. $\frac{3}{2}\frac{m{v}_0^2}{x_0^2}$
• D. $\frac{2}{3}\frac{m{v}_0^2}{x_0^2}$

Answer 1

Answer: (D)

Using conservation of linear momentum, we have
$m{v}_0=mv+2mv$
$\Rightarrow v=\frac{v_0}{3}$
Using conservation of energy, we have
$\frac{1}{2}m{v}_0^2=\frac{1}{2}k{x}_0^2+\frac{1}{2}(3m)v^2$
where $x_0$ is compression in the string.
$\therefore m{v}_0^2=k{x}_0^2+(3m)\frac{v_0^2}{g}$
$\Rightarrow k{x}_0^2=m{v}_0^2-\frac{m{v}_0^2}{3}$
$\Rightarrow k{x}_0^2=\frac{2m{v}_0^2}{3}$
$\therefore k=\frac{2m{v}_0^2}{3x_0^2}$