Question

Two blocks A and B each of mass m are connected by a massless spring of natural length L and spring constant k. The blocks are initially resting on a smooth horizontal floor with the spring as its natural length, as shown in figure.

A third identical block C, also of mass m moves on the floor with a speed v along the line joining A and B collides with A, elastically, then

• A. The kinetic energy of AB system at maximum compression of the spring is zero.
• B. The kinetic energy of the AB system at maximum of the spring is .
• C. The maximum compression of the spring is v .
• D. The maximum compression of the spring is v

Answer 1

Answer: (B), (D)

The correct options are
B The kinetic energy of the AB system at maximum of the spring is .
D The maximum compression of the spring is v


Due to equal mass, C transfers its total momentum to A. therefore A has the velocity v. At max. compression the velocity of both the
bodies are equal say, v'. Conservation of linear momentum yields
$mv=(m+m)v^{{}^{\prime }}\Rightarrow v^{{}^{\prime }}=\left(\frac{v}{2}\right)=\frac{1}{2}(m+m)v^{{}^{\prime }2}=\frac{1}{2}\left(2m\right)\frac{v^2}{4}=\frac{m{v}^2}{4}\frac{1}{2}k{x}^2=\Delta KE=\frac{m{v}^2}{2}-\frac{m{v}^2}{4}\Rightarrow x=\left(\sqrt{\frac{m}{2k}}\right)v.$