Question:
A ball \(A\) collides with another identical ball \(B\) which is at rest. After collision, if the velocity of ball \(B\) becomes two times the final velocity of ball \(A\), then the coefficient of restitution is
A ball \(A\) collides with another identical ball \(B\) which is at rest. After collision, if the velocity of ball \(B\) becomes two times the final velocity of ball \(A\), then the coefficient of restitution is
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For collisions of identical masses, first apply conservation of momentum to relate the final velocities. Then use
\[
e=\frac{\text{speed of separation}}{\text{speed of approach}}.
\]
This usually gives the answer in one or two steps.
Updated On: Jul 29, 2026
- \[ \frac13 \]
- \[ \frac12 \]
- \[ \frac14 \]
- \[ \frac16 \]
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The Correct Option is A
Solution and Explanation
Concept:
For a one-dimensional collision,
\[
e=
\frac{\text{Relative velocity of separation}}
{\text{Relative velocity of approach}}.
\]
Also, linear momentum is conserved.
Step 1: Apply conservation of momentum. Let the initial velocity of ball \(A\) be \[ u. \] Since ball \(B\) is at rest, \[ u_B=0. \] Let final velocities be \[ v_A=v, \qquad v_B=2v. \] Since the masses are identical, \[ mu=m v_A+m v_B. \] \[ u=v+2v. \] \[ u=3v. \] \[ v=\frac{u}{3}. \] Therefore, \[ v_A=\frac{u}{3}, \qquad v_B=\frac{2u}{3}. \]
Step 2: Use the definition of coefficient of restitution. \[ e = \frac{v_B-v_A}{u_A-u_B}. \] Substituting, \[ e = \frac{\frac{2u}{3}-\frac{u}{3}} {u-0}. \] \[ = \frac{\frac{u}{3}}{u}. \] \[ = \frac13. \] Therefore, \[ \boxed{e=\frac13} \] \[ \boxed{\text{Answer = (A)}} \]
Step 1: Apply conservation of momentum. Let the initial velocity of ball \(A\) be \[ u. \] Since ball \(B\) is at rest, \[ u_B=0. \] Let final velocities be \[ v_A=v, \qquad v_B=2v. \] Since the masses are identical, \[ mu=m v_A+m v_B. \] \[ u=v+2v. \] \[ u=3v. \] \[ v=\frac{u}{3}. \] Therefore, \[ v_A=\frac{u}{3}, \qquad v_B=\frac{2u}{3}. \]
Step 2: Use the definition of coefficient of restitution. \[ e = \frac{v_B-v_A}{u_A-u_B}. \] Substituting, \[ e = \frac{\frac{2u}{3}-\frac{u}{3}} {u-0}. \] \[ = \frac{\frac{u}{3}}{u}. \] \[ = \frac13. \] Therefore, \[ \boxed{e=\frac13} \] \[ \boxed{\text{Answer = (A)}} \]
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