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Chapter 7: Problem 75

A 60.0 -kg astronaut inside a 7.00 -m-long space capsule of mass \(500 . \mathrm{kg}\) is floating weightlessly on one end of the capsule. He kicks off the wall at a velocity of \(3.50 \mathrm{~m} / \mathrm{s}\) toward the other end of the capsule. How long does it take the astronaut to reach the far wall?

Short Answer

Expert verified
Answer: It takes approximately 1.79 seconds for the astronaut to reach the far wall of the space capsule.

Step by step solution

01

Calculate the initial and final momentum of the astronaut and the capsule

Before the astronaut kicks off, both the astronaut and the capsule are at rest, so their initial momenta are zero. When the astronaut kicks off with a velocity of \(3.50\,\text{m/s}\), the final momentum of the astronaut (\(P_{\text{astronaut}}\)) is given by: \(P_{\text{astronaut}}= m_{\text{astronaut}} \cdot v_{\text{astronaut}} = 60.0\,\text{kg}\cdot 3.50\,\text{m/s}= 210\,\text{kg路m/s}\) According to the conservation of momentum principle, the final momentum of the capsule (\(P_{\text{capsule}}\)) should be equal and opposite to that of the astronaut. \(P_{\text{capsule}} = -P_{\text{astronaut}} = -210\,\text{kg路m/s}\)
02

Calculate the final velocity of the capsule

To find the final velocity of the capsule (\(v_{\text{capsule}}\)), we can use the following equation: \(v_{\text{capsule}} = \dfrac{P_{\text{capsule}}}{m_{\text{capsule}}} = \dfrac{-210\,\text{kg路m/s}}{500\,\text{kg}} = -0.42\,\text{m/s}\)
03

Calculate the relative velocity between the astronaut and the capsule

The relative velocity between the astronaut and the capsule (\(v_{\text{relative}}\)) is given by: \(v_{\text{relative}} = v_{\text{astronaut}} - v_{\text{capsule}} = 3.50\,\text{m/s} - (-0.42\,\text{m/s}) = 3.92\,\text{m/s}\)
04

Calculate the time taken by the astronaut to reach the far wall

The distance between the astronaut and the far wall of the capsule is 7.00 m: distance = 7.00 m To find the time taken by the astronaut to reach the far wall, we can use the following equation: time = \(\dfrac{\text{distance}}{v_{\text{relative}}} = \dfrac{7.00\,\text{m}}{3.92\,\text{m/s}} = 1.79\,\text{s}\) Therefore, it takes approximately 1.79 seconds for the astronaut to reach the far wall of the space capsule.

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Most popular questions from this chapter

The value of the momentum for a system is the same at a later time as at an earlier time if there are no a) collisions between particles within the system. b) inelastic collisions between particles within the system. c) changes of momentum of individual particles within the system. d) internal forces acting between particles within the system. e) external forces acting on particles of the system.

Although they don't have mass, photons-traveling at the speed of light-have momentum. Space travel experts have thought of capitalizing on this fact by constructing solar sails-large sheets of material that would work by reflecting photons. Since the momentum of the photon would be reversed, an impulse would be exerted on it by the solar sail, and-by Newton's Third Law-an impulse would also be exerted on the sail, providing a force. In space near the Earth, about \(3.84 \cdot 10^{21}\) photons are incident per square meter per second. On average, the momentum of each photon is \(1.30 \cdot 10^{-27} \mathrm{~kg} \mathrm{~m} / \mathrm{s}\). For a \(1000 .-\mathrm{kg}\) spaceship starting from rest and attached to a square sail \(20.0 \mathrm{~m}\) wide, how fast could the ship be moving after 1 hour? One week? One month? How long would it take the ship to attain a speed of \(8000 . \mathrm{m} / \mathrm{s}\), roughly the speed of the space shuttle in orbit?

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