Mass of astronaut gas?

Question

An astronaut in a space suit is motionless in outer space. The propulsion unit strapped to her back ejects some gas with a velocity of 50 m/s. The astronaut and space suit after the gas is ejected is 120 kg, the mass of the gas ejected is???

please help thanks!! best answer gets 10 points!

Answer 1

From the law of conservation of momentum we know that the final momentum of the astronaut + the momentum of the gas must equal the initial momentum of the astronaut-gas combination, which in this case equals zero since the system is initially at rest.
P_initial = P_final = 0

Momentum = mass * velocity

The initial momentum of the system is zero since the astronaut-gas combination are at rest with respect to whatever frame of reference the question is using (I really dislike how the question just says the astronaut is at rest in space without stating a frame of reference).

The final momentum is the sum of the final momentums of the gas and the astronaut.

The final momentum of the astronaut can be found as,
P_f_a = m_a * v_f_a
The final momentum of the gas can be found as,
P_f_g = m_g * v_f_g

We are told that, in the end, the astronaut has a mass of 120 kg (m_a) and we are also told that the gas is ejected with a speed of 50 m/s (v_f_g). We do not know the final speed of the astronaut and we are trying to solve for the mass of the ejected gas.
We do, however know that the final momentum equals the initial momentum equals zero.

P_f_a + P_f_g = m_a * v_f_a + m_g * v_f_g = 0,
Solving fro the mass of the ejected gas,
m_g = -(m_a * v_f_a) / (v_f_g)
Plugging in,
m_g = -(120 kg * v_f_a) / (50 m/s)
Once you find the final velocity of the astronaut, you can solve this equation for the mass of the gas.

*it is important to note that the velocity of the astronaut will be in the opposite direction as the velocity of the gas and thus have the opposite sign when you plug into the equation. I simply assumed the velocity of the gas to be in the “positive” direction and thus the velocity of the astronaut to be in the “negative” direction…but it could easily be the other way around without affected the results, just make sure you remember to include the negative sign when you plug in.

Answer 2

Sorry, not enough data to answer this question fully. But here is a partial answer.

From Newton's third law, the kinetic energy of the gas and the kinetic energy of the astronaut (after the venting) must be the same. Since E = 1/2 m v^2, we can write:
1/2 120 Va^2 = 1/2 Mg 50^2

where Va is the velocity of the astronaut, and Mg is the mass of the gas. This simplifies to:

Mg = 6/125 Va^2

... which is as close as we can come without knowing the astronaut's velocity.

Answer 3

The guy above me is right, not enough info but the momentum of the gas and the momentum of the astronaut will be the same but just opposite in direction.
That means:

m_astro*v_astro=m_gas*v_gas
120*v_astro=m_gas*50
m_gas=12/5*v_astro