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Chapter 5: Problem 68

A ball \(A\) of mass \(m\) moving with velocity \(u\), collides head on with another ball \(B\) of the same mass at rest. If the co-efficient of restitution between balls is \(e\), the ratio of the final and initial velocities of ball \(A\) will be (A) \(\frac{1+e}{1-e}\) (B) \(\frac{1+e}{2}\) (C) \(1-e\) (D) \(\frac{1-e}{2}\)

Short Answer

Expert verified
The short answer to the given problem is: The ratio of the final and initial velocities of ball A is \(\frac{1-e}{1+e}\), which corresponds to option (A).

Step by step solution

01

Conservation of momentum before and after collision

Before the collision, Ball A has momentum \(m \times u\), and Ball B has zero momentum, as it is at rest. After the collision, Ball A has momentum \(m \times v_{A}\), and Ball B has momentum \(m \times v_{B}\). By the conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision. Therefore, we can write the equation: \(m \times u = m \times v_{A} + m \times v_{B}\) Divide by m to obtain: \(u = v_{A} + v_{B}\)
02

Coefficient of restitution equation

The coefficient of restitution \(e\) is defined as the ratio of the relative speed of separation after the collision to the relative speed of approach before the collision: \(e = \frac{v_{B} - v_{A}}{u - 0}\) From this equation, we can express \(v_{B}\) in terms of \(v_{A}\) and \(u\): \(v_{B} = e \times u + v_{A}\)
03

Substitute \(v_B\) into the conservation of momentum equation

Now, substitute the expression for \(v_B\) from Step 2 into the conservation of momentum equation from Step 1: \(u = v_{A} + (e \times u + v_{A})\) Solve for the final velocity of ball A, \(v_A\): \(v_A = \frac{u(1 - e)}{1 + e}\)
04

Find the ratio of final and initial velocities of ball A

We need to find the ratio of the final velocity of ball A to its initial velocity, which can be expressed as \(\frac{v_A}{u}\). \(\frac{v_A}{u} = \frac{\frac{u(1 - e)}{1 + e}}{u}\) Simplifying, we get: \(\frac{v_A}{u} = \frac{1-e}{1+e}\) Therefore, the ratio of the final and initial velocities of ball A is \(\frac{1-e}{1+e}\), which corresponds to option (A).

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