Conservation of momentum astronaut problem

[pureouchies4717]

Homework Statement

An astronaut in his space suit and with a propulsion unit (empty of its gas propellant) strapped to his back has a mass of 143 kg. During a space-walk, the unit, which has been filled with propellant gas, ejects some gas with a velocity of +30.9 m/s. As a result, the astronaut recoils with a velocity of -0.265 m/s. After the gas is ejected, the mass of the astronaut (now wearing a partially empty propulsion unit) is 167 kg. What percentage of the gas propellant in the completely filled propulsion unit was depleted?

Homework Equations

p=mv

The Attempt at a Solution

-.265(astronaut + gas) = 30.9 (gas)
-.265(143 + x) = 30.9x
-37.846 -.265x = 30.9x
31.165x = -37.846
x = 1.214 kg


i don't really know where to go from here... can someone please help?

 

[Doc Al]

You need to distinguish between the portion of gas left in the propulsion unit and the portion of gas expelled. You should be able to immediately figure out the portion in the container from the data given. Use the conservation of momentum equation to find the portion expelled.

The Attempt at a Solution

-.265(astronaut + gas) = 30.9 (gas)

Correct this equation. The "gas" on the left is the portion left in the unit, which should not be an unknown. And get rid of that minus sign on the left.

 

[Moetasim]

First, it is important to note that this problem is a conservation of momentum problem, as stated in the title. This means that the total momentum before and after the gas is ejected must be equal. We can use the equation p=mv, where p is momentum, m is mass, and v is velocity, to solve this problem.

Before the gas is ejected, the total momentum is equal to the momentum of the astronaut and the propulsion unit combined. We can represent this as:

p_before = (mass of astronaut + mass of propulsion unit) * velocity of astronaut and propulsion unit

p_before = (143 kg + 0 kg) * (-0.265 m/s)

p_before = -37.845 kg*m/s

After the gas is ejected, the total momentum is equal to the momentum of the astronaut and the partially empty propulsion unit. We can represent this as:

p_after = (mass of astronaut + mass of partially empty propulsion unit) * velocity of astronaut and partially empty propulsion unit

p_after = (143 kg + 24 kg) * (-0.265 m/s)

p_after = -37.845 kg*m/s

Since the total momentum before and after the gas is ejected must be equal, we can set these two equations equal to each other and solve for the mass of the partially empty propulsion unit:

-37.845 = (143 + 24) * (-0.265)

-37.845 = 167 * (-0.265)

-37.845 = -44.2555

44.2555 = 37.845

This means that the mass of the partially empty propulsion unit is 24 kg. To find the percentage of gas propellant depleted, we can use the equation:

% depleted = (mass of gas propellant initially - mass of gas propellant remaining) / mass of gas propellant initially * 100

% depleted = (24 kg - 0 kg) / 24 kg * 100

% depleted = 100%

Therefore, 100% of the gas propellant in the completely filled propulsion unit was depleted. This makes sense, as the astronaut recoiled with a velocity of -0.265 m/s, indicating that all of the gas was ejected and there was no remaining propellant to provide additional momentum.