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
$ mv_0 = mv + 2 mv $
$ \Rightarrow v = \frac{ v_0 }{ 3} $
Using conservation of energy, we have
$ \frac{1}{2} mv_0^2 = \frac{1}{2} kx_0^2 + \frac{1}{2} ( 3m)v^2 $
where $ x_0 $ is compression in the string.
$ \therefore mv_0^2 = kx_0^2 + ( 3 m ) \frac{ v_0^2 }{ g } $
$ \Rightarrow kx_0^2 = mv_0^2 - \frac{ mv_0^2 }{ 3} $
$ \Rightarrow kx_0^2 = \frac{ 2 mv_0^2 }{ 3 } $
$ \therefore k = \frac{ 2 mv_0^2 }{ 3 x_0^2 } $