Answer:
Approximately [tex]0.88\; {\rm m \cdot s^{-1}}[/tex] to the right (assuming that both astronauts were originally stationary.)
Explanation:
If an object of mass [tex]m[/tex] is moving at a velocity of [tex]v[/tex], the momentum [tex]p[/tex] of that object would be [tex]p = m\, v[/tex].
Since momentum of this system (of the astronauts) conserved:
[tex]\begin{aligned} &(\text{Total Final Momentum}) \\ &= (\text{Total Initial Momentum})\end{aligned}[/tex].
Assuming that both astronauts were originally stationary. The total initial momentum of the two astronauts would be [tex]0[/tex] since the velocity of both astronauts was [tex]0\![/tex].
Therefore:
[tex]\begin{aligned} &(\text{Total Final Momentum}) \\ &= (\text{Total Initial Momentum})\\ &= 0\end{aligned}[/tex].
The final momentum of the first astronaut ([tex]m = 64\; {\rm kg}[/tex], [tex]v = 0.8\; {\rm m\cdot s^{-1}}[/tex] to the left) would be [tex]p_{1} = m\, v = 64\; {\rm kg} \times 0.8\; {\rm m\cdot s^{-1}} = 51.2\; {\rm kg \cdot m \cdot s^{-1}}[/tex] to the left.
Let [tex]p_{2}[/tex] denote the momentum of the astronaut in question. The total final momentum of the two astronauts, combined, would be [tex](p_{1} + p_{2})[/tex].
[tex]\begin{aligned} & p_{1} + p_{2} \\ &= (\text{Total Final Momentum}) \\ &= (\text{Total Initial Momentum})\\ &= 0\end{aligned}[/tex].
Hence, [tex]p_{2} = (-p_{1})[/tex]. In other words, the final momentum of the astronaut in question is the opposite of that of the first astronaut. Since momentum is a vector quantity, the momentum of the two astronauts magnitude ([tex]51.2\; {\rm kg \cdot m \cdot s^{-1}}[/tex]) but opposite in direction (to the right versus to the left.)
Rearrange the equation [tex]p = m\, v[/tex] to obtain an expression for velocity in terms of momentum and mass: [tex]v = (p / m)[/tex].
[tex]\begin{aligned}v &= \frac{p}{m} \\ &= \frac{51.2\; {\rm kg \cdot m \cdot s^{-1}}}{64\; {\rm kg}} && \genfrac{}{}{0}{}{(\text{to the right})}{} \\ &\approx 0.88\; {\rm m\cdot s^{-1}} && (\text{to the right})\end{aligned}[/tex].
Hence, the velocity of the astronaut in question ([tex]m = 58.2\; {\rm kg}[/tex]) would be [tex]0.88\; {\rm m \cdot s^{-1}}[/tex] to the right.
