A sphere of mass m moving with a constant velocity v hits...

A sphere of mass $m$ moving with a constant velocity $v$ hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution, then the ratio of the velocities of the two spheres after collision will be (a) $\left(\frac{e+1}{e-1}\right)$ (b) $\left(\frac{e-1}{e+1}\right)$ (c) $\left(\frac{1-e}{e+1}\right)$ (d) $\left(\frac{1+e}{e-1}\right)$

A sphere of mass $m$ moving with a constant velocity $v$ hits another stationary sphere of the same mass. If $e$ is the coefficient of restitution, then the ratio of the velocities of the two spheres after collision will be

(a) $\left(\frac{e+1}{e-1}\right)$
(b) $\left(\frac{e-1}{e+1}\right)$
(c) $\left(\frac{1-e}{e+1}\right)$
(d) $\left(\frac{1+e}{e-1}\right)$

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Key Concepts

Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a system remains constant if no external forces act upon it. In collision problems, this principle is used to relate the masses and velocities before and after the collision, forming one of the key equations necessary to determine the outcomes of the collision.

Collision Dynamics in Equal Mass Systems

When two objects of equal mass collide, the analysis simplifies because the symmetry allows for specific algebraic relationships between the final velocities derived from both momentum conservation and the coefficient of restitution. This is particularly useful in determining ratios of velocities in head-on collisions where one of the masses might be initially stationary.

Coefficient of Restitution

The coefficient of restitution (e) measures the elasticity of a collision and is defined as the ratio of the relative speed after collision to the relative speed before collision, taken along the line of impact. It indicates how much kinetic energy is conserved in the collision, with e = 1 representing a perfectly elastic collision and e = 0 representing a perfectly inelastic one.

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