astronaut and gas?

Question

An astronaut in space suit is motionless in outer space. The propulsion unit is strapped to her back ejects some gas with a velocity of 50 m/s. The astronaut recoils with a velocity of 1.0 m/s. If the mass of the astronaut and space suit after gas is ejected is 120 kg., the mass of the gas ejected is???

please help thanks!

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 that the astronaut has a final speed of 1 m/s. We are also told that the gas is ejected with a speed of 50 m/s (v_f_g).

P_f_a + P_f_g = m_a * v_f_a + m_g * v_f_g = 0,
Solving for the mass of the ejected gas,
m_g = -(m_a * v_f_a) / (v_f_g)

Remember that the velocity of the astronaut is in the opposite direction as the velocity of the ejected gas. If we call the speed of the astronaut +1 m/s, then the velocity of the gas is -50 m/s.

Plugging in,
m_g = -(120 kg * 1 m/s) / (-50 m/s)

The mass of the ejected gas comes out to be: 2.4 kg.

Answer 2

Conservation of momentum
120 kg * 1m/s = x * 50m/s