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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: \(P_{\text{astronaut}}= m_{\text{astronaut}} \cdot v_{\text{astronaut}} = 60.0\,\text{kg}\cdot 3.50\,\text{m/s}= 210\,\text{kg·m/s}\) 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. \(P_{\text{capsule}} = -P_{\text{astronaut}} = -210\,\text{kg·m/s}\)
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: \(v_{\text{capsule}} = \dfrac{P_{\text{capsule}}}{m_{\text{capsule}}} = \dfrac{-210\,\text{kg·m/s}}{500\,\text{kg}} = -0.42\,\text{m/s}\)
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

Tennis champion Venus Williams is capable of serving a tennis ball at around 127 mph. a) Assuming that her racquet is in contact with the 57.0 -g ball for \(0.250 \mathrm{~s}\), what is the average force of the racquet on the ball? b) What average force would an opponent's racquet have to exert in order to return Williams's serve at a speed of \(50.0 \mathrm{mph}\), assuming that the opponent's racquet is also in contact with the ball for 0.250 s?

Two Sumo wrestlers are involved in an inelastic collision. The first wrestler, Hakurazan, has a mass of \(135 \mathrm{~kg}\) and moves forward along the positive \(x\) -direction at a speed of \(3.5 \mathrm{~m} / \mathrm{s}\). The second wrestler, Toyohibiki, has a mass of \(173 \mathrm{~kg}\) and moves straight toward Hakurazan at a speed of \(3.0 \mathrm{~m} / \mathrm{s} .\) Immediately after the collision, Hakurazan is deflected to his right by \(35^{\circ}\) (see the figure). In the collision, \(10 \%\) of the wrestlers' initial total kinetic energy is lost. What is the angle at which Toyohibiki is moving immediately after the collision?

A billiard ball of mass \(m=0.250 \mathrm{~kg}\) hits the cushion of a billiard table at an angle of \(\theta_{1}=60.0^{\circ}\) at a speed of \(v_{1}=27.0 \mathrm{~m} / \mathrm{s}\) It bounces off at an angle of \(\theta_{2}=71.0^{\circ}\) and a speed of \(v_{2}=10.0 \mathrm{~m} / \mathrm{s}\). a) What is the magnitude of the change in momentum of the billiard ball? b) In which direction does the change of momentum vector point?

A hockey puck with mass \(0.250 \mathrm{~kg}\) traveling along the blue line (a blue-colored straight line on the ice in a hockey rink) at \(1.50 \mathrm{~m} / \mathrm{s}\) strikes a stationary puck with the same mass. The first puck exits the collision in a direction that is \(30.0^{\circ}\) away from the blue line at a speed of \(0.750 \mathrm{~m} / \mathrm{s}\) (see the figure). What is the direction and magnitude of the velocity of the second puck after the collision? Is this an elastic collision?

When a \(99.5-\mathrm{g}\) slice of bread is inserted into a toaster, the toaster's ejection spring is compressed by \(7.50 \mathrm{~cm}\). When the toaster ejects the toasted slice, the slice reaches a height \(3.0 \mathrm{~cm}\) above its starting position. What is the average force that the ejection spring exerts on the toast? What is the time over which the ejection spring pushes on the toast?
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