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 +30. 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 find the percentage of gas propellant ejected from the propulsion unit, we need to calculate the change in momentum of the astronaut before and after the gas is ejected. We can then use this information to determine the mass of the gas propellant.
The initial momentum of the astronaut before the gas is ejected is given by:
Initial momentum = Mass of astronaut x Initial velocity of astronaut
The final momentum of the astronaut after the gas is ejected is given by:
Final momentum = Mass of astronaut x Final velocity of astronaut
The change in momentum is calculated by subtracting the initial momentum from the final momentum:
Change in momentum = Final momentum - Initial momentum
Since momentum is conserved, the change in momentum of the astronaut is equal to the change in momentum of the gas propellant. Therefore, we can write:
Change in momentum = Mass of gas propellant x Gas propellant velocity
We are given the masses of the astronaut before and after the gas is ejected, as well as the velocities of the gas propellant and the astronaut. We can now set up the equation:
Mass of astronaut x Initial velocity of astronaut = (Mass of astronaut - Mass of gas propellant) x Final velocity of astronaut
Plugging in the given values:
146 kg x 0 m/s = (165 kg - Mass of gas propellant) x (-0.39 m/s)
Simplifying the equation:
0 = -0.39(165 kg - Mass of gas propellant)
We can now solve for the mass of the gas propellant:
165 kg - Mass of gas propellant = 0
Mass of gas propellant = 165 kg
Therefore, the entire gas propellant was ejected from the propulsion unit.
To find the percentage of gas propellant ejected, we can use the equation:
Percentage of gas propellant ejected = (Mass of gas propellant ejected / Initial mass of gas propellant) x 100%
Since the initial mass of gas propellant was 165 kg and the entire gas propellant was ejected, the percentage of gas propellant ejected is:
Percentage of gas propellant ejected = (165 kg / 165 kg) x 100%
Percentage of gas propellant ejected = 100%
Therefore, 100% of the gas propellant was ejected from the completely filled propulsion unit.
To solve this problem, we need to apply the principle of conservation of momentum. The total momentum before the gas is ejected is equal to the total momentum after the gas is ejected.
Let's break down the problem step by step:
1. Determine the initial momentum:
The initial momentum is equal to the mass of the astronaut plus the mass of the propulsion unit (both completely filled) multiplied by their initial velocity, which is zero since the astronaut is at rest.
Initial momentum = (mass of astronaut + mass of propulsion unit) * initial velocity
= (146 kg + mass of propulsion unit) * 0
2. Determine the momentum after the gas is ejected:
The momentum after the gas is ejected is equal to the mass of the astronaut (now including the partially empty propulsion unit) multiplied by the final velocity of the astronaut.
Final momentum = (mass of astronaut + mass of partially empty propulsion unit) * final velocity
= 165 kg * -0.39 m/s
3. Apply conservation of momentum:
According to the principle of conservation of momentum, the initial momentum is equal to the final momentum.
(146 kg + mass of propulsion unit) * 0 = 165 kg * -0.39 m/s
4. Solve for the mass of the propulsion unit:
Cross-multiply and solve for the mass of the propulsion unit:
146 kg + mass of propulsion unit = (165 kg * -0.39 m/s) / 0
mass of propulsion unit = (165 kg * -0.39 m/s) / 0 - 146 kg
5. Calculate the percentage of the gas propellant ejected:
The amount of gas propellant ejected can be determined by subtracting the mass of the propulsion unit after the ejection from the mass of the propulsion unit when it was completely filled.
Percentage ejected = (mass of propulsion unit when completely filled - mass of propulsion unit after ejection) / mass of propulsion unit when completely filled * 100
Substitute the calculated value for the mass of the propulsion unit:
Percentage ejected = (mass of propulsion unit when completely filled - ((165 kg * -0.39 m/s) / 0 - 146 kg)) / mass of propulsion unit when completely filled * 100
Now you can substitute the given values and calculate the percentage of the gas propellant ejected.
momentum conserved.
assuming this is an impulse..
M=146
m=mass of gas emitted
m*30=(165+m).39
m(29.61)=165*.39
m=you do it.
percentage=m/(165+m-149)