Two blocks A and B, each of mass m are connected by a...

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 at its natural length, as shown in figure. A third block $C$, also of mass $m$, moves on the floor with a speed $v$ along the line joining $A$ and $B$ and collides with A. Then : (a) the kinetic energy of the $A+B$ system at maximum compression of the spring is zero (b) the kinetic energy of $A+B$ system, at maximum compression of the spring is $\frac{m v^{2}}{4}$ (c) the maximum compression of the spring is $v \sqrt{\left(\frac{m}{k}\right)}$ (d) none of the above

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 at its natural length, as shown in figure.
A third block $C$, also of mass $m$, moves on the floor with a speed $v$ along the line joining $A$ and $B$ and collides with
A. Then :
(a) the kinetic energy of the $A+B$ system at maximum compression of the spring is zero
(b) the kinetic energy of $A+B$ system, at maximum compression of the spring is $\frac{m v^{2}}{4}$
(c) the maximum compression of the spring is $v \sqrt{\left(\frac{m}{k}\right)}$
(d) none of the above

Key Concepts

Collision Dynamics Involving Springs

Collisions that involve springs introduce additional complexities compared to simple rigid-body collisions. The massless spring transmits forces between the colliding bodies, leading to oscillatory motion and energy exchange between kinetic and potential forms. Understanding these dynamics is vital to determine the system’s behavior at instances such as the point of maximum spring compression.

Maximum Compression of a Spring

Maximum compression occurs when the relative motion between the objects connected by the spring momentarily reaches zero. At this instant, the system’s kinetic energy is at a minimum and the maximum amount of energy is temporarily stored as elastic potential energy in the spring, enabling the calculation of the compression distance using spring parameters.

Conservation of Linear Momentum

In collision problems, the total momentum of the system remains constant if no external forces act. This principle allows one to relate the pre?collision and post-collision velocities of the interacting bodies, which is crucial for determining the motion of the blocks involved after the impact.

Energy Conservation in Systems with Springs

When springs are involved, mechanical energy is exchanged between kinetic energy and potential energy stored in the spring. At specific points, such as maximum compression, the kinetic energy is minimized and the stored potential energy is maximized. This interplay is key to analyzing the energy distribution during and after collisions.

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