Two blocks A and B are attached to the two ends of a spring...

Two blocks $\mathrm{A}$ and $\mathrm{B}$ are attached to the two ends of a spring having length $\mathrm{L}$ and force constant $\mathrm{k}$ on a horizontal surface. Initially the system is in equilibrium. Now a third block having same mass $\mathrm{m}$, moving with velocity v collides with block A. In this situation .......... (A) During maximum contraction of the spring, the kinetic energy of the system $\mathrm{A}-\mathrm{B}$ will be zero. (B) During maximum contraction of the spring, the kinetic energy of the system $\mathrm{A}-\mathrm{B}$ will be $\mathrm{mV}^{2} / 4$ (C) Maximum contraction of the spring is $\mathrm{V} \sqrt{(\mathrm{m} / \mathrm{k})}$ (D) Maximum contraction of the spring is $\mathrm{V} \sqrt{(2 \mathrm{~m} / \mathrm{k})}$

Two blocks $\mathrm{A}$ and $\mathrm{B}$ are attached to the two ends of a spring having length $\mathrm{L}$ and force constant $\mathrm{k}$ on a horizontal surface. Initially the system is in equilibrium. Now a third block having same mass $\mathrm{m}$, moving with velocity v collides with block A. In this situation ..........
(A) During maximum contraction of the spring, the kinetic energy of the system $\mathrm{A}-\mathrm{B}$ will be zero.
(B) During maximum contraction of the spring, the kinetic energy of the system $\mathrm{A}-\mathrm{B}$ will be $\mathrm{mV}^{2} / 4$
(C) Maximum contraction of the spring is $\mathrm{V} \sqrt{(\mathrm{m} / \mathrm{k})}$
(D) Maximum contraction of the spring is $\mathrm{V} \sqrt{(2 \mathrm{~m} / \mathrm{k})}$

Key Concepts

Center-of-Mass Motion

Analyzing the system from the center-of-mass frame simplifies the problem of understanding the combined motion. At maximum spring compression, the common velocity of the masses reflects the center-of-mass velocity. This concept is critical because it distinguishes the overall translational movement of the system from the internal energy changes due to the spring's compression and expansion.

Conservation of Linear Momentum

When analyzing collisions, the principle of linear momentum conservation is crucial. This principle states that in an isolated system with no external forces, the total momentum before and after an event remains constant. It helps in determining the common velocity of interacting bodies after impact, even when energy transformations occur within the system.

Energy Transformation and Conservation

Even though momentum is conserved in a collision, kinetic energy might not be, particularly in inelastic collisions. Within systems involving springs, energy is continuously exchanged between kinetic energy and elastic potential energy stored in the spring. This energy conversion process is central to understanding the dynamics after the collision and during oscillatory motion.

Spring-Mass Oscillations and Maximum Compression

In a spring-mass system, maximum compression occurs at the instant when the two attached masses momentarily move together with the same velocity, thus minimizing their relative kinetic energy. At this point, a significant amount of the system’s energy is stored as elastic potential energy in the spring, which then drives the oscillatory motion as it gets converted back into kinetic energy.

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