Problem 75 - University Physics with Modern Physics

Question

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?

Step-by-Step Solution

Answer

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

Step 1: 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. \(P_{\text{capsule}} = -P_{\text{astronaut}} = -210\,\text{kg·m/s}\)

Step 2: Calculate the final velocity of the capsule

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

Step 3: 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: \(v_{\text{relative}} = v_{\text{astronaut}} - v_{\text{capsule}} = 3.50\,\text{m/s} - (-0.42\,\text{m/s}) = 3.92\,\text{m/s}\)

Step 4: 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.