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?

검증된 단계별 안내

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 \( v_2 \).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event must equal the total momentum after the event, provided no external forces act on it. In this scenario, the momentum of the train cars before they couple must equal the momentum of the combined train after they couple.

Momentum Calculation

Momentum (p) is calculated as the product of mass (m) and velocity (v), expressed as p = mv. For the train cars, we need to consider the mass of each car and their respective velocities to determine the total momentum before and after the coupling events.

Inelastic Collision

An inelastic collision occurs when two objects collide and couple together, resulting in a loss of kinetic energy but conservation of momentum. In this problem, the coupling of the train cars and the stationary fifth car is an inelastic collision, which affects the final speed of the combined train.