Before the collision, the total momentum of the system is:
$$P_i = m_1v_1 + m_2(0)$$
where:
- \(P_i\) is the total initial momentum
- \(m_1\) is the mass of the moving boxcar
- \(v_1\) is the velocity of the moving boxcar
- \(m_2\) is the mass of the boxcar at rest
After the collision, the two boxcars move together with a common velocity \(v\). The total momentum of the system after the collision is:
$$P_f = (m_1 + m_2)v$$
Since the total momentum of the system must be conserved, we have:
$$P_i = P_f$$
$$m_1v_1 + m_2(0) = (m_1 + m_2)v$$
Solving for \(v\), we get:
$$v = \frac{m_1v_1}{m_1 + m_2}$$
This expression gives us the velocity of the two boxcars after the collision. The combined momentum of the two boxcars after the collision is:
$$P = (m_1 + m_2)v = \frac{m_1m_2v_1}{m_1 + m_2}$$
Therefore, the combined momentum of the two boxcars after the collision is equal to the momentum of the moving boxcar before the collision, divided by the sum of the masses of the two boxcars.
