A block A of mass m connected with a spring of force...

A block $A$ of mass $m$ connected with a spring of force constant $k$ is executing SHM. The position $(x)$ and time $(t)$ equation of the block is $x=x_{0}+a \sin \omega t$. An identical block $B$ moving towards negative $x$-axis with velocity $v_{0}$ collides elastically with block $A$ at time $t=0$. Then (A) Displacement time equation of $A$ after collision will be $x=x_{0}-v_{0} \sqrt{\frac{m}{k}} \sin \omega t$. (B) Displacement time equation of $A$ after collision will be $x=x_{0}+v_{0} \sqrt{\frac{m}{k}} \sin \omega t$. (C) Velocity of $B$ just after collision will be $a \omega$ towards positive $x$-direction. (D) Velocity of $B$ just after collision will be $v_{0}$ towards positive $x$-direction.

A block $A$ of mass $m$ connected with a spring of force constant $k$ is executing SHM. The position $(x)$ and time $(t)$ equation of the block is $x=x_{0}+a \sin \omega t$. An identical block $B$ moving towards negative $x$-axis with velocity $v_{0}$ collides elastically with block $A$ at time $t=0$. Then
(A) Displacement time equation of $A$ after collision will be $x=x_{0}-v_{0} \sqrt{\frac{m}{k}} \sin \omega t$.
(B) Displacement time equation of $A$ after collision will be $x=x_{0}+v_{0} \sqrt{\frac{m}{k}} \sin \omega t$.
(C) Velocity of $B$ just after collision will be $a \omega$ towards positive $x$-direction.
(D) Velocity of $B$ just after collision will be $v_{0}$ towards positive $x$-direction.

Key Concepts

Simple Harmonic Motion (SHM)

SHM is a type of periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position, typically described by a sine or cosine function. The parameters such as amplitude, angular frequency (? = ?(k/m)), and phase determine the motion characteristics, and this concept is fundamental when analyzing any system that exhibits oscillations, such as a mass attached to a spring.

Elastic Collision

An elastic collision is one in which both momentum and kinetic energy are conserved. When two objects collide elastically, they exchange momentum and energy without any loss due to heat or deformation. This principle is key in analyzing interactions between objects such as the colliding blocks, ensuring the post-collision velocities can be determined by applying conservation laws.

Conservation of Momentum and Energy

In mechanical collisions, especially elastic ones, the total momentum and total kinetic energy of the system remain constant. These conservation laws provide the basis for calculating the velocities of colliding bodies after impact. They are essential for determining the outcome of interactions between an oscillator and an incoming body, influencing subsequent motion equations.

Phase Shift in Oscillatory Motion due to Impulsive Forces

When an impulsive force, such as that resulting from a collision, is applied to an oscillator, the amplitude and phase of the SHM can change abruptly. This phase shift alters the displacement-time equation of the oscillator, reflecting changes in the direction of motion and the system's response immediately following the collision. Understanding this helps in formulating the correct post-collision motion equation.

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