Two equal spheres of mass m are in contact on a smooth...

Two equal spheres of mass $m$ are in contact on a smooth horizontal table. A third identical sphere impinges symmetrically on them and reduces to rest. Then (a) Coefficient of restitution is $e=\frac{2}{3}$. (b) Loss of kinetic energy $\frac{1}{6} m u^{2}$ where $u$ is velocity before impact. (c) After the collision, velocity of equal mass sphere is $\frac{u}{\sqrt{3}}$. (d) Loss of kinetic energy $\frac{1}{3} m u^{2}$.

Two equal spheres of mass $m$ are in contact on a smooth horizontal table. A third identical sphere impinges symmetrically on them and reduces to rest. Then
(a) Coefficient of restitution is $e=\frac{2}{3}$.
(b) Loss of kinetic energy $\frac{1}{6} m u^{2}$ where $u$ is velocity before impact.
(c) After the collision, velocity of equal mass sphere is $\frac{u}{\sqrt{3}}$.
(d) Loss of kinetic energy $\frac{1}{3} m u^{2}$.

Key Concepts

Symmetry in Collision Analysis

Symmetry in collision analysis allows simplification of complex problems by reducing the number of independent variables. When multiple identical bodies are involved and the collision is arranged symmetrically, one can often assume equal distribution of momentum or energy among the bodies. This aids in simplifying calculations of final velocities and energy losses.

Kinetic Energy Loss in Collisions

In many collision scenarios, especially inelastic ones, some of the system’s kinetic energy is converted into other forms such as heat or sound, resulting in energy loss. Quantifying this loss involves comparing the initial and final kinetic energies, which is essential in understanding the nature of the collision and the efficiency of energy transfer.

Momentum Conservation

Momentum conservation is a fundamental principle stating that the total momentum of an isolated system remains constant if no external forces act upon it. In collision problems, this principle is critical to relate the velocities of objects before and after impact, regardless of whether the collision is elastic or inelastic.

Coefficient of Restitution

The coefficient of restitution is a measure of how elastic a collision is. It is defined as the ratio of the relative speed after collision to the relative speed before collision along the line of impact. This concept helps in determining how much kinetic energy is conserved in a collision, with a value of 1 indicating a perfectly elastic collision and lower values indicating inelastic behavior.

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