Ch 11: Impulse and Momentum

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 11: Impulse and Momentum

Problem 62

Chapter 11, Problem 62

A 30 ton rail car and a 90 ton rail car, initially at rest, are connected together with a giant but massless compressed spring between them. When released, the 30 ton car is pushed away at a speed of 4.0 m/s relative to the 90 ton car. What is the speed of the 30 ton car relative to the ground?

Verified step by step guidance

Step 1: Start by identifying the principle of conservation of momentum. Since the system is initially at rest, the total momentum of the system before and after the spring is released must remain zero. This means the momentum of the 30-ton car and the 90-ton car must cancel each other out.

Step 2: Write the momentum equation for the system. Let the speed of the 30-ton car relative to the ground be \( v_{30} \) and the speed of the 90-ton car relative to the ground be \( v_{90} \). The momentum equation is: \( m_{30} v_{30} + m_{90} v_{90} = 0 \), where \( m_{30} = 30 \ \text{tons} \) and \( m_{90} = 90 \ \text{tons} \).

Step 3: Use the given relative velocity between the two cars. The problem states that the 30-ton car moves at 4.0 m/s relative to the 90-ton car. This means \( v_{30} - v_{90} = 4.0 \ \text{m/s} \).

Step 4: Solve the system of equations. You now have two equations: \( m_{30} v_{30} + m_{90} v_{90} = 0 \) and \( v_{30} - v_{90} = 4.0 \). Substitute \( v_{90} = v_{30} - 4.0 \) into the momentum equation and solve for \( v_{30} \).

Step 5: Once you solve for \( v_{30} \), you will have the speed of the 30-ton car relative to the ground. Remember to check your solution to ensure it satisfies both the momentum conservation equation and the relative velocity condition.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 initial momentum of the system (both rail cars at rest) is zero, which means the momentum after the spring is released must also sum to zero, allowing us to relate the velocities of the two cars.

Relative Velocity

Relative velocity is the velocity of one object as observed from another object. In this problem, the speed of the 30 ton car is given relative to the 90 ton car. To find its speed relative to the ground, we must consider the motion of both cars and how their velocities interact, particularly since they are moving apart due to the spring's release.

Newton's Third Law of Motion

Newton's Third Law states that for every action, there is an equal and opposite reaction. When the spring pushes the 30 ton car away, it exerts a force on it, while simultaneously exerting an equal and opposite force on the 90 ton car. This interaction is crucial for understanding how the velocities of both cars are determined after the spring is released.

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