MHT CET · Physics · Center of Mass Momentum and Collision

A sphere of mass $m$ moving with velocity $v$ collides head-on another sphere of same mass which is at rest. The ratio of final velocity of the second sphere to the initial velocity of the first sphere is ( $e$ is the coefficient of restitution and collision is inelastic)

A $\frac{e - 1}{2}$
B $\frac{e}{2}$
C $\frac{e + 1}{2}$
D $e$

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Answer & Solution

Correct Answer

(C) $\frac{e + 1}{2}$

Step-by-step Solution

Detailed explanation

m u + 0 = m u_{1} + m u_{2}

υ_{1} + υ_{2} = υ

....(i)

e = \frac{υ_{2} - υ_{1}}{υ - 0}

e υ = υ_{2} - υ_{1}

..(ii)
Add equation (i) & equation (ii)

e υ + υ = 2 υ_{2}

υ_{2} = \frac{υ (e + 1)}{2}

\frac{υ_{2}}{υ} = \frac{e + 1}{2}

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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A sphere of mass \(m\) moving with velocity \(v\) collides head-on another sphere of same mass which is at rest. The ratio of final velocity of the second sphere to the initial velocity of the first sphere is ( \(e\) is the coefficient of restitution and collision is inelastic)

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Step-by-step Solution

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