Two carts on a frictionless track are moving toward each other. Cart A has a mass of 2 kg and is moving to the right with a speed of 3 m/s. Cart B has a mass of 3 kg and is moving to the left with a speed of 2 m/s. What is the velocity of the center of mass of this system?

Core Concept

The velocity of the center of mass is given by v_cm = (m_A*v_A + m_B*v_B)/(m_A + m_B). Taking right as positive: v_cm = (2 kg × 3 m/s + 3 kg × (-2 m/s))/(2 kg + 3 kg) = (6 kg·m/s - 6 kg·m/s)/5 kg = 0 kg·m/s/5 kg = 0 m/s. However, if we consider the direction of motion, cart A has more momentum in the positive direction (6 kg·m/s) than cart B has in the negative direction (6 kg·m/s), so the center of mass should move to the right. The correct calculation is v_cm = (2 kg × 3 m/s + 3 kg × (-2 m/s))/(2 kg + 3 kg) = (6 - 6)/5 = 0. But this suggests the center of mass is stationary, which contradicts our intuition. Let me recalculate: v_cm = (m_A*v_A + m_B*v_B)/(m_A + m_B) = (2 kg × 3 m/s + 3 kg × (-2 m/s))/(5 kg) = (6 - 6)/5 = 0 m/s. Actually, the center of mass is stationary because the total momentum is zero.