An astronaut in his space suit and with a propulsion unit (empty of its gas propellant) strapped to his back has a mass of 146 kg. The astronaut begins a space walk at rest, with a completely filled propulsion unit. During the space walk, the unit ejects some gas with a velocity of +43 m/s. As a result, the astronaut recoils with a velocity of -0.39 m/s. After the gas is ejected, the mass of the astronaut (now wearing a partially empty propulsion unit) is 165 kg. What percentage of the gas propellant was ejected from the completely filled propulsion unit?
To solve this problem, we need to make use of the principle of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, provided there are no external forces acting on the system.
In this case, we have an astronaut and a propulsion unit as the system. We know the initial mass of the system (146 kg), its velocity after the ejection of gas (-0.39 m/s), and the final mass of the system (165 kg).
Let's break down the steps to find the percentage of gas propellant ejected:
1. Calculate the initial momentum:
Initial Momentum = Mass × Velocity = 146 kg × 0 m/s (since the astronaut is at rest initially) = 0 kg m/s
2. Calculate the final momentum:
Final Momentum = Mass × Velocity = 165 kg × -0.39 m/s = -64.35 kg m/s (Note: The negative sign indicates the direction of the momentum)
3. Calculate the momentum change:
Momentum Change = Final Momentum - Initial Momentum
4. The momentum change is equal to the momentum of the ejected gas:
Momentum of Ejected Gas = Momentum Change
5. Calculate the mass of the ejected gas:
Mass of Ejected Gas = Momentum of Ejected Gas / Velocity of Ejected Gas
6. Calculate the initial mass of the propulsion unit:
Initial Mass of Propulsion Unit = Mass of System - Mass of Ejected Gas
7. Calculate the percentage of gas propellant ejected:
Percentage of Ejected Gas = (Mass of Ejected Gas / Initial Mass of Propulsion Unit) × 100
Let's substitute the given values into the equations to find the answer step by step:
1. Initial Momentum = 0 kg m/s
2. Final Momentum = 165 kg × -0.39 m/s = -64.35 kg m/s
3. Momentum Change = -64.35 kg m/s - 0 kg m/s = -64.35 kg m/s
4. Momentum of Ejected Gas = -64.35 kg m/s
5. Mass of Ejected Gas = (-64.35 kg m/s) / (43 m/s) = -1.498 kg (Note: Since velocity is positive, the mass will be negative to reflect the opposite direction of momentum)
6. Initial Mass of Propulsion Unit = 146 kg - (-1.498 kg) = 147.498 kg
7. Percentage of Ejected Gas = (-1.498 kg / 147.498 kg) × 100 = -1.02%
Therefore, approximately 1.02% of the gas propellant was ejected from the completely filled propulsion unit.
It is important to note that in this calculation, we assumed that the gas was ejected in the opposite direction of the astronaut's initial velocity.