91Ó°ÊÓ

In physics, momentum is a crucial concept. It describes the quantity of motion an object has and is calculated by multiplying the object's mass by its velocity.
Momentum can be represented by the formula:
\[ p = m imes v \]
where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. In our space scenario, both the astronaut and the ejected gas have momentum.
The principle of conservation of momentum states that, in a closed system with no external forces, the total momentum before and after an event must be equal. This forms the basis for solving problems like this space walk exercise and helps us understand the motion resulting when the gas is ejected.

A space walk, also known as an Extravehicular Activity (EVA), involves an astronaut moving outside their spacecraft in space. It requires precise equipment such as a propulsion unit, which can adjust an astronaut's position in space without touching anything.
In the problem, the astronaut starts at rest in space, meaning they aren't moving relative to the spacecraft or other nearby objects.
Regaining motion in the vacuum of space happens through reactions, like the ejection of gas from the propulsion unit. This exemplifies Newton's third law, where every action has an equal and opposite reaction. Here, conserving momentum allows the astronaut to control movement when the gas is expelled.

To solve the problem about the ejected gas, accurately calculating mass is essential. We start by understanding the mass of the astronaut, both with a full and partially empty propulsion unit.
Initially, the astronaut along with a completely filled propulsion unit has a total mass of 165 kg. When the propulsion unit is empty, this decreases to 146 kg. Subtract these to find the mass of the fully filled unit's gas, which is 19 kg.
Through the momentum equation and the recoil speed of the astronaut, we can reverse calculate the mass of the ejected gas, aiding in further analysis of the system's dynamics.

As the propulsion unit releases gas, it alters the astronaut's momentum, which is crucial for movement in space. The gas is ejected at a speed of +32 m/s, and calculating the exact mass it represents is a matter of using conservation laws.
Starting with the total system's stationary condition, the negative and positive momenta from the astronaut and the gas must cancel out.
By applying the formula \( m_g v_g = -m_a v_a \), and knowing the astronaut's velocity change and the system's initial conditions, we find the ejected gas's mass is approximately 2.011 kg.
In percentage terms, this ejected mass represents approximately 10.58% of the initial gas. Understanding this process highlights the precision required in mass and velocity estimations for successful space maneuvers.