An astronaut in his space suit and with a propulsion unit (empty of its gas propellant) strapped to his back has a mass of 134 kg. During a space-walk, the unit, which has been completely filled with propellant gas, ejects some gas with a velocity of +32 m/s. As a result, the astronaut recoils with a velocity of -0.32 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 in the completely filled propulsion unit was depleted?
To find the percentage of gas propellant depleted in the propulsion unit, we need to calculate the change in momentum for the astronaut and the propulsion unit.
First, we find the initial momentum of the astronaut and the propulsion unit. Given that the astronaut recoils with a velocity of -0.32 m/s and has a mass of 134 kg, the initial momentum of the astronaut can be calculated as:
Initial Momentum of Astronaut = Mass of Astronaut × Velocity of Astronaut
= 134 kg × (-0.32 m/s)
= -42.88 kg·m/s
Next, we find the final momentum of the astronaut and the partially empty propulsion unit. Given that the mass of the astronaut (now wearing the partially empty propulsion unit) is 165 kg, and assuming the propulsion unit still carries the same amount of gas propellant, we can calculate the final momentum of the system as:
Final Momentum of Astronaut + Final Momentum of Propulsion Unit = 0
Let's represent the mass of the partially empty propulsion unit as "x" (in kg). We know that the gas propellant is not depleted from the propulsion unit, so the change in mass only occurs in the astronaut. Thus, the mass of the gas propellant (in kg) can be written as:
Mass of Gas Propellant = 134 kg - x kg
Using the conservation of momentum, we can write the equation:
Initial Momentum of Astronaut + Initial Momentum of Propulsion Unit
= Final Momentum of Astronaut + Final Momentum of Propulsion Unit
-42.88 kg·m/s + 0 kg·m/s = 165 kg × Vf + (134 kg - x kg) × 0 m/s
Simplifying the equation, we have:
-42.88 kg·m/s = 165 kg × Vf
Now, let's solve for Vf:
Vf = -42.88 kg·m/s / 165 kg
≈ -0.26 m/s
Therefore, the final velocity of the astronaut is approximately -0.26 m/s.
Now, to find the percentage of gas propellant depleted, we need to calculate the change in mass of the propulsion unit. Given that the initial mass of the propulsion unit is "x" kg, and the final mass is 165 kg, we can find the change in mass as:
Change in Mass = Initial Mass - Final Mass
= x kg - 165 kg
The percentage of gas propellant depleted is given by:
Percentage of Gas Propellant Depleted = (Change in Mass / Initial Mass) × 100%
Let's substitute the values into the equation and solve for the percentage:
Percentage of Gas Propellant Depleted = ((x kg - 165 kg) / x kg) × 100%
Please provide the initial mass of the propulsion unit (x) to determine the percentage of gas propellant depleted.