91Ó°ÊÓ

When dealing with space, especially in zero-gravity environments like that of a space station, understanding the concept of velocity is crucial. Here, we focus on how the astronaut's velocity is affected when they push off an object, like a satellite. Due to the conservation of momentum, any push the astronaut exerts will also exert an equal and opposite reaction back on them. This principle is dictated by Newton's third law of motion.

The conservation of momentum is expressed mathematically as:

• The total initial momentum before the push is zero, because both the astronaut and satellite start from rest.
• After the push, the total momentum must still equal zero.

Using this concept, we can derive the astronaut's velocity (\(v_a\)). The equation: \[m_a v_a + m_s v_s = 0\] can be rearranged to solve for \(v_a\), representing the astronaut's movement towards the station as they push away from the satellite. The negative sign indicates the opposite direction to the satellite's movement.

The satellite movement is an integral part of understanding the dynamics within this scenario. When the astronaut pushes the satellite, it moves with a certain velocity in the opposite direction. This movement is quantified with the given speed of the satellite, here being \(0.14 ext{ m/s}\). The push affects both objects but in opposite directions, as dictated by the conservation of momentum.

This scenario demonstrates an isolated system, where no external forces act upon the satellite and the astronaut apart from their mutual interaction. This means:

• The satellite's speed \((v_s = 0.14 ext{ m/s})\) is directly connected to the way the astronaut moves.
• The calculations assume an ideal condition where external influences like air resistance are absent.

Through understanding the satellite's movement, students can connect these fundamental physics concepts to real-world applications, like how astronauts maneuver in space during missions.

Finally, calculating the initial distance from the astronaut to the space station involves understanding both velocity and time. In this scenario, although the astronaut moves towards the station upon pushing the satellite, the task is to calculate how far away they were initially, before starting to move.

The formula for calculating distance is:

• Using \( ext{distance} = ext{velocity} imes ext{time} \), where time here is 7.5 seconds it takes the astronaut to reach the station.
• The astronaut's velocity, calculated as approximately \(-1.826 ext{ m/s}\), is inserted into the equation.

Carrying out this multiplication gives us the initial distance:\[1.826 imes 7.5 = 13.695 ext{ meters}\] This distance reflects the starting point relative to the station after pushing the satellite.