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When we talk about astronaut dynamics, we are delving into the realm of how objects, like astronauts and their tools, move and interact in the unique conditions of space. Unlike movements on Earth, space dynamics are influenced mainly by Newton's laws of motion without the interference of gravity. In the scenario of our astronaut working on a space station, the interaction between her and the tool she throws is a classic example of a system free from external forces. The space environment allows us to observe pure momentum conservation. Inertia, the tendency of an object to resist changes to its state of motion, plays a significant role here. Both the astronaut and tool were initially at rest, floating together. When she exerted a force by throwing the tool, it created an equal and opposite reaction according to Newton's third law of motion. This reaction is what sets the astronaut in motion.

The concept of momentum is essential when analyzing the movement of objects in space. Momentum is defined as the product of an object's mass and its velocity, represented mathematically as: \[ p = m imes v \] In our astronaut’s example, the principle of conservation of momentum applies. This principle dictates that in an isolated system, the total momentum before and after an event must be equal. Initially, our entire system comprising the astronaut and the tool has zero momentum because both were at rest. After she throws the tool, the tool gains momentum. The conservation of momentum ensures that the total momentum remains zero, as expressed in the equation:\[ m_1 v_1 + m_2 v_2 = 0 \] Here, the negative momentum of the astronaut balances the positive momentum of the tool, keeping the overall system momentum constant.

In these types of physics problems, calculating the velocity of an astronaut or any object after an interaction is crucial for understanding the outcome of momentum changes. The task is straightforward if we apply the conservation principles and basic algebra.To find the astronaut's velocity: 1. Recognize the initial zero momentum.2. Use the momentum equation: \( m_1 v_1 + m_2 v_2 = 0 \).3. Rearrange to solve for the unknown velocity:\[ v_1 = \frac{-m_2 \, v_2}{m_1} \] Substituting in the given values: \[ v_1 = \frac{-2.25 \, \times \, 3.20}{68.5} \approx -0.105 \, \text{m/s} \]The negative sign indicates the direction is opposite to the tool’s motion. This calculated velocity reflects the astronaut's slow drift backward as a result of throwing the tool. Understanding the direction and magnitude of her movement helps reinforce core concepts in astronaut dynamics and momentum conservation.