Question

A block of mass m moving with a velocity v hits a light spring of stiffness K attached rigidly to a stationary sledge of mass M. Neglecting friction between all contacting surfaces, the maximum compression of the spring is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

The velocity of the combinational at the time of maximum compression of the spring = v = $\frac{m{v}_0}{M+m}$ obtained by conserving the momentum of the system.

The $\Delta$ $K.E_{system})=-[\frac{1}{2}$ m ${v_0}^2$ - $\frac{1}{2}$ (M + m) $v^2$]

By putting v = $\frac{m{v}_0}{M+m}$, we obtain $\Delta$$K.E_{system}$ = $\frac{Mm{v_0}^2}{2(M+m)}$

$\Delta$ $P.E_{system}$ = $\frac{1}{2}$ k $x^2$ where x = maximum compression of the spring.

$\Rightarrow$ ($\Delta$ PE + $\Delta KE{)}_{system}$ = 0

$\Rightarrow$ $\frac{1}{2}$ k $x^2$ - $\frac{Mm{v_0}^2}{2(M+m)}$ = 0

$\Rightarrow$ x = [ $\sqrt{\frac{Mn}{(M+m)k}}$ ] $v_0$