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The impulse-momentum theorem is a fundamental idea in physics that explains how the momentum of an object changes when a force is applied over a period of time. This concept is crucial in our understanding of how the astronaut and the space capsule interact.

When the astronaut pushes away from the capsule, she experiences a change in momentum. This happens because her velocity changes from 0 to 2.50 m/s. According to the theorem, this change in momentum equals the impulse applied by the force over time. The formula for this is \[ F \times \Delta t = m \Delta v \]where:

• \( F \) is the force applied
• \( \Delta t \) is the time period over which the force acts
• \( m \Delta v \) is the change in momentum

In our scenario, both the astronaut and the capsule experience this force, but in opposite directions, in line with Newton's Third Law. By calculating the impulse, we can determine the average force each applies to the other over the given time frame of 0.600 s. This gives us valuable insights into how the astronaut and capsule set their motion.

Kinetic energy is the energy of motion. For any moving object, we calculate it using the formula:

\[ KE = \frac{1}{2}mv^2 \]Here, \( m \) is the mass and \( v \) is the velocity of the object. In our exercise, both the astronaut and the capsule have kinetic energy once they start moving after the push.

For the astronaut, the mass is 125 kg, moving at a speed of 2.50 m/s. Plugging into the formula, we calculate:\[ KE_{\text{astronaut}} = \frac{1}{2} \times 125 \times (2.50)^2 = 390.625 \text{ J} \]
The space capsule, significantly heavier at 1900 kg, moves slower; its velocity is -0.1645 m/s due to the push:\[ KE_{\text{capsule}} = \frac{1}{2} \times 1900 \times (0.1645)^2 = 25.70 \text{ J} \]
The differences here illustrate that a small object with high velocity can possess more kinetic energy than a large object moving slowly. These calculations help us understand how energy is split between the two objects after the push.

Newton's Third Law of Motion is pivotal in understanding interactions like the push-off between the astronaut and the capsule. It states: "For every action, there is an equal and opposite reaction."

This means whenever the astronaut applies a force on the space capsule, the capsule applies an equal force back on her, but in the opposite direction.

In practical terms, this is why the astronaut gains speed in one direction, while the capsule moves in the opposite direction.

• The force exerted by the astronaut on the capsule causes the capsule to accelerate in the opposite direction of her push.
• Similarly, the force by the capsule ensures that the astronaut moves forward at a calculated 2.50 m/s.

This law not only helps explain the motions post-impact but also ensures that the total momentum remains conserved. Thus, the astronaut and capsule's combined effects don't just highlight their individual actions but underline a balanced interaction essential in zero-gravity environments.