A body \(A\) moving with momentum \(P\) collides one-dimensionally with another stationary body \(B\) of…

A body \(A\) moving with momentum \(P\) collides one-dimensionally with another stationary body \(B\) of same mass. During impact, \(A\) gives impulse \(J\) to \(B\). Then which of the following is/are correct? (a) The total momentum of \(A\) and \(B\) is \(P\) before and after impact and \((P-J)\) during the impact. (b) During the impact, \(B\) gives impulse of magnitude \(J\) to \(A\). (c) The coefficient of restitution is \(\left[\frac{2 J}{P}-1\right]\). (d) The coefficient of restitution is \(\left[\frac{2 J}{P}+1\right]\).
  1. Only (a) is correct.
  2. (a) and (c) are correct.
  3. (b) and (c) are correct.
  4. Only (c) is correct.

Solution

According to the question, a collision between two bodies (\(A\) and \(B\)) is shown in the figure below,
So, the coefficient of restitution \(\begin{aligned} e & =\frac{\text { Relative velocity of separation }}{\text { Relative velocity of approach }} \\ \Rightarrow \quad e & =\frac{\frac{J-p}{m}+\frac{J}{m}}{\frac{p}{m}-0}=\frac{2 J}{P}-1 \quad\left(\because v=\frac{m v}{m}=\frac{p}{m}\right) \end{aligned}\) During the impact, \(B\) gives impulse \(J_B=p+J-p=J\) Hence, the statement (b) and (c) are correct.

Asked in: AP EAMCET 2019 (23 Apr Shift 1)

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