Question

8.3 elastic and inelastic collisions

1. two people, who have the same mass, throw two different objects at the same velocity. if the first object is heavier than the second, compare the velocities gained by the two people as a result of recoil.

a. the first person will gain more velocity as a result of recoil.
b. the second person will gain more velocity as a result of recoil.
c. both people will gain the same velocity as a result of recoil.
d. the velocity of both people will be zero as a result of recoil.

Explanation:

Step1: Apply the law of conservation of momentum

The law of conservation of momentum states that \(m_1v_1 + m_2v_2=m_1v_1'+m_2v_2'\). Initially, assume the two - people - object systems are at rest (\(v = 0\)). Let the mass of each person be \(M\), the mass of the first object be \(m_1\) and its velocity after throw be \(v_1\), the mass of the second object be \(m_2\) and its velocity after throw be \(v_2\), and the recoil velocities of the two people be \(V_1\) and \(V_2\) respectively.
From conservation of momentum \(0 = m_1v_1+MV_1\) and \(0 = m_2v_2+MV_2\). So, \(V_1=-\frac{m_1v_1}{M}\) and \(V_2 =-\frac{m_2v_2}{M}\). Given \(v_1 = v_2\) (same throwing velocity) and \(m_1>m_2\) (first object is heavier).

Step2: Compare the magnitudes of recoil velocities

Take the magnitudes \(\vert V_1\vert=\frac{m_1\vert v_1\vert}{M}\) and \(\vert V_2\vert=\frac{m_2\vert v_2\vert}{M}\). Since \(m_1 > m_2\) and \(\vert v_1\vert=\vert v_2\vert\), we have \(\vert V_1\vert>\vert V_2\vert\).

Answer:

a. The first person will gain more velocity as a result of recoil.