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Chapter 15: Problem 86

Two smooth billiard balls \(A\) and \(B\) each have a mass of \(200 \mathrm{~g}\). If \(A\) strikes \(B\) with a velocity \(\left(v_{A}\right)_{1}=1.5 \mathrm{~m} / \mathrm{s}\) as shown, determine their final velocities just after collision. Ball \(B\) is originally at rest and the coefficient of restitution is \(e=0.85 .\) Neglect the size of each ball.

Short Answer

Expert verified
The final velocities of balls A and B can be calculated by solving the set of equations that emerge from applying the conservation of momentum and the coefficient of restitution. By executing these steps, ball A's and ball B's final velocities can be determined.

Step by step solution

01

Applying conservation of momentum

We know the initial velocity of ball A is \((v_A)_1 = 1.5 m/s\), and that of ball B is \((v_B)_1 = 0\), since it's at rest. We also know the mass of both balls is \(m = 200g = 0.2 kg\). The initial momentum of the system (before collision) would be \((m_A*(v_A)_1) + (m_B*(v_B)_1) = m*(v_A)_1 + 0 = 0.2*1.5 = 0.3 kg.m/s\). Let's denote the final velocities of ball A and B after collision as \((v_A)_2\) and \((v_B)_2\) respectively. The final momentum of the system (after collision) would be \((m_A*(v_A)_2) + (m_B*(v_B)_2) = m*(v_A)_2 + m*(v_B)_2\). Since there are no external forces acting on the system, the total momentum before and after the collision will remain the same. So the momentum equation will be \(0.2*(v_A)_2 + 0.2*(v_B)_2 = 0.3\).
02

Applying the coefficient of restitution

The coefficient of restitution 'e' is given by \((v_B)_2 - (v_A)_2)/((v_A)_1 - (v_B)_1)\), where the numerator is the difference in velocities after collision and the denominator is the difference in velocities before collision. This is given as 0.85 in the problem. After substituting the given values, we obtain: 0.85 = \((v_B)_2 - (v_A)_2)/(1.5 - 0)\). Simplifying this equation yields: \((v_B)_2 - (v_A)_2 = 1.275\).
03

Solve the equations

We now have a system of two equations from Step 1 and Step 2, and two unknowns (\((v_A)_2\) and \((v_B)_2\)). Solving the equations, \((v_A)_2\) and \((v_B)_2\) can be calculated. These will be the final velocities of ball A and B respectively just after the collision.

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Most popular questions from this chapter

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