Question

Three identical train cars, coupled together, are rolling east at speed v0. A fourth car traveling east at 2v0 catches up with the three and couples to make a four-car train. A moment later, the train cars hit a fifth car that was at rest on the tracks, and it couples to make a five-car train. What is the speed of the five-car train?

Accepted Answer

Step 1: Identify the principle of conservation of momentum, which states that the total momentum before a collision is equal to the total momentum after the collision, provided no external forces act on the system. Step 2: Calculate the total momentum of the three coupled train cars and the fourth car before they couple. The momentum of the three cars is $$ p_1 = 3m v_0 $$, and the momentum of the fourth car is $$ p_2 = m (2v_0) . $$Add these to find the total initial momentum before coupling: $$ p_{initial1} = 3m v_0 + m (2v_0) . $$Step 3: After the fourth car couples with the three cars, the total mass becomes $$ 4m . $$Use conservation of momentum to find the velocity $$ v_1  of $$the four-car train: $$ p_{initial1} = p_{final1} $$, where $$ p_{final1} = (4m) v_1 . $$Solve for $$ v_1 . $$Step 4: Next, calculate the total momentum of the four-car train and the fifth car before they couple. The momentum of the four-car train is $$ p_3 = (4m) v_1 $$, and the momentum of the fifth car (at rest) is $$ p_4 = 0 . $$Add these to find the total initial momentum before coupling: $$ p_{initial2} = (4m) v_1 + 0 . $$Step 5: After the fifth car couples with the four-car train, the total mass becomes $$ 5m . $$Use conservation of momentum to find the velocity $$ v_2  of $$the five-car train: $$ p_{initial2} = p_{final2} $$, where $$ p_{final2} = (5m) v_2 . $$Solve for $$ v_2 .$$

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Key Concepts

Conservation of Momentum

Momentum Calculation

Inelastic Collision