The total momentum of the system (astronaut+gas) must be conserved.
We can assume the astronaut is still before the gas starts to be ejected, therefore its speed is zero and its momentum is zero as well.
After the gas starts to be ejected, the total momentum of the system is:
[tex]p=m_A v_A + m_G v_G[/tex]
where [tex]m_A=50 kg[/tex] is the mass of the astronaut, [tex]v_A[/tex] is the speed of the astronaut, [tex]m_G=100 g=0.1 kg[/tex] is the mass of the gas and
[tex]v_G=50 m/s[/tex] is the speed of the gas.
Since the momentum must be conserved, and the initial momentum was zero, then it must be [tex]p=0[/tex]. Using this information, we can find the value of [tex]v_A[/tex], the speed of the astronaut:
[tex]0=m_Av_A + m_G v_G[/tex]
[tex]v_A=- \frac{m_Gv_G}{m_A}=- \frac{(0.1 kg)(50 m/s)}{50 kg}=-0.1 m/s [/tex]
where the negative sign means that the astronaut starts to move in the opposite direction of the ejected gas.
