Question

Two blocks $A$ and $B$ of mass $m$ and $2m$ are connected by a massless spring of force constant $k$. They are placed on a smooth horizontal plane. Spring is stretched by an amount $x$ and then released. The relative velocity of the blocks when the spring comes to its natural length is

• A. $\left(\sqrt{\frac{3k}{2m}}\right)x$
• B. $\left(\sqrt{\frac{2k}{3m}}\right)x$
• C. $\sqrt{\frac{2kx}{m}}$
• D. $\sqrt{\frac{3km}{2x}}$

Answer 1

The correct option is A $\left(\sqrt{\frac{3k}{2m}}\right)x$

From conservation of mechanical energy
$\frac{1}{2}k{x}^2=\frac{1}{2}\mu v_r^2$ ... (1)
Here, $\mu =$ reduced mass of the blocks
$=\frac{\left(m\right)\left(2m\right)}{m+2m}=\frac{2}{3}m$
and $v_r=$ relative velocity of the two.
Substituting in Equation (1), we get
$k{x}^2=\frac{2}{3}m{v}_r^2$
$\therefore v_r=\left(\sqrt{\frac{3k}{2m}}\right)x$