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Answer :
The question involves a physics concept related to momentum conservation and relative motion in space, where we calculate the resulting speed of a space capsule after an astronaut pushes off from it. By applying the law of conservation of momentum, and knowing the mass and speed of the astronaut, we can determine the speed at which the capsule will move in the opposite direction.
In this scenario, the astronaut is pushing off from the space capsule to move in the opposite direction. According to Newton's third law of motion, for every action, there is an equal and opposite reaction.
1. Initially, the astronaut and the capsule are at rest in space. When the astronaut pushes off from the capsule, she exerts a force on the capsule in one direction. As a result, the capsule exerts an equal but opposite force on the astronaut, propelling her in the opposite direction.
2. The total momentum of the system (astronaut + capsule) remains constant because no external forces are acting on the system. Therefore, the momentum of the astronaut after pushing off must be equal in magnitude but opposite in direction to the momentum of the capsule.
3. By applying the principle of conservation of momentum, you can calculate the speed of the astronaut after pushing off from the capsule. The momentum of the astronaut before pushing off is zero (as the astronaut is at rest). After pushing off, the combined momentum of the astronaut and the capsule is redistributed between them in opposite directions.
4. To find the speed of the astronaut, you can use the equation:
[tex]\[ m_{\text{astronaut}} \times v_{\text{astronaut}} = m_{\text{capsule}} \times v_{\text{capsule}} \][/tex]
where:
[tex]\( m_{\text{astronaut}}[/tex] = [tex]115 \, \text{kg} \) (mass of the astronaut)[/tex],
[tex]\( v_{\text{capsule}}[/tex] = [tex]0 \, \text{m/s} \) (initial velocity of the capsule),[/tex]
[tex]\( m_{\text{capsule}}[/tex] = [tex]1800 \, \text{kg} \) (mass of the capsule),[/tex]
[tex]\( v_{\text{astronaut}}[/tex] = [tex]2.10 \, \text{m/s} \) (final velocity of the astronaut).[/tex]
5. Solving the equation for the final velocity of the astronaut [tex](\( v_{\text{astronaut}} \))[/tex], you can determine the speed acquired by the astronaut after pushing off from the capsule.
By understanding and applying the principles of conservation of momentum and Newton's third law of motion, you can calculate the speed of the astronaut after pushing off from the capsule in space.
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