Understanding Elastic Collisions: Relative Speed and Conservation Principles

The statement that in an elastic collision, the relative speed of the bodies after the collision is equal to the relative speed before the collision, is indeed true. This phenomenon is a direct consequence of the conservation principles of momentum and kinetic energy.

Principles at Play

In an elastic collision, both the conservation of momentum and the conservation of kinetic energy hold true. One of the key characteristics of such collisions is that the relative speed of the two bodies before the collision is equal to their relative speed after the collision, but in the opposite direction. This can be mathematically represented as:

Where v_1 and v_2 are the velocities of the two bodies before the collision, and u_1 and u_2 are their velocities after the collision.

In Single-Dimensional Collisions

In single-dimensional elastic collisions, the relative speed of the bodies after the collision is equal to the relative speed before the collision, but the direction is reversed. For example, if the two bodies approach each other at 2 m/s before the collision, they will depart from each other at 2 m/s after the collision, in the opposite direction. This principle can be intuitively grasped by considering the motion in the center-of-mass reference frame:

Center-of-Mass Frame of Reference

Consider two bodies of arbitrary mass approaching each other with no other forces acting on them. In a reference frame based on the center-of-mass of the two bodies prior to the collision, the momenta of the two bodies must be equal and opposite since no net external force acts on the center-of-mass. After the elastic collision, in this reference frame, the two bodies must rebound with a relative velocity that is identical in magnitude but opposite in direction to the relative velocity before the collision.

The invariance of the relative velocity under any transformation to another reference frame is a fundamental property of elastic collisions. This means that regardless of the observer's viewpoint, the relative speed before and after the collision remains the same.

Elastic vs. Inelastic Collisions

Elastic collisions, as described, contrast with inelastic collisions, where the internal kinetic energy is not conserved. In inelastic collisions, the objects may coalesce or otherwise lose some of their kinetic energy to heat or deformation.

Thus, the key to understanding the relative speed in elastic collisions lies in the principles of conservation of momentum and kinetic energy, and the invariance of relative velocity in the center-of-mass frame.

Conclusion

The statement about the relative speed in elastic collisions is indeed true and can be explained by the fundamental principles of physics. Understanding this concept is crucial for anyone studying dynamics, mechanics, or related fields in physics.