Ch. 09 - Linear Momentum

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 09 - Linear Momentum

Problem 70a

Chapter 9, Problem 70a

A huge balloon and its gondola, of mass M, are in the air and stationary with respect to the ground. A passenger, of mass m, then climbs out and slides down a rope with speed v, measured with respect to the balloon. With what speed and direction (relative to Earth) does the balloon then move? What happens if the passenger stops?

Verified step by step guidance

Start by applying the principle of conservation of momentum. Since the system (balloon + gondola + passenger) is initially stationary with respect to the ground, the total momentum of the system is zero before the passenger starts moving.

Define the variables: Let the mass of the balloon and gondola be M, the mass of the passenger be m, and the speed of the passenger relative to the balloon be v. Let the speed of the balloon relative to the Earth after the passenger starts moving be V (to be determined).

Write the equation for conservation of momentum. The total momentum of the system after the passenger starts moving must still be zero. The momentum of the passenger relative to the Earth is m(v - V), and the momentum of the balloon relative to the Earth is MV. Set up the equation: \( m(v - V) + MV = 0 \).

Solve for V (the speed of the balloon relative to the Earth). Rearrange the equation: \( V = \frac{mv}{M + m} \). The direction of the balloon's motion will be opposite to the direction of the passenger's motion relative to the balloon.

If the passenger stops moving (relative to the balloon), the system becomes stationary again with respect to the Earth. This is because the passenger and the balloon are no longer exchanging momentum, and the system's total momentum remains zero.

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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 balloon and passenger system is initially stationary, so the total momentum is zero. When the passenger slides down, the momentum of the passenger moving downward must be balanced by an equal and opposite momentum change in the balloon, causing it to move upward.

Relative Velocity

Relative velocity refers to the velocity of one object as observed from another object. In this case, the passenger's speed v is measured relative to the balloon. To find the balloon's speed relative to the Earth, one must consider both the passenger's downward velocity and the balloon's response to maintain momentum, which will result in the balloon moving in the opposite direction to the passenger's motion.

Effect of Stopping

When the passenger stops sliding down the rope, their velocity relative to the balloon becomes zero. This sudden stop will affect the momentum of the system. The balloon will experience a change in momentum due to the passenger's change in motion, potentially causing it to move slightly in the opposite direction to the passenger's previous motion, depending on the dynamics of the system at that moment.

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