Answer :
Sure! Let's go through the solution step-by-step for the astronaut problem using conservation of momentum.
1. Identify the System and Initial Conditions:
- We have an astronaut in space, initially at rest, with a mass [tex]\( m_{\text{astronaut}} = 84 \)[/tex] kg.
- The astronaut fires a thruster that expels gas with a mass [tex]\( m_{\text{gas}} = 0.035 \)[/tex] kg at a velocity [tex]\( v_{\text{gas}} = 875 \, \text{m/s} \)[/tex].
2. Apply Conservation of Momentum:
- Since there are no external forces, momentum is conserved. \\
- Initially, both the astronaut and the gas are at rest, so the total initial momentum is zero.
3. Write the Momentum Equation:
- Final momentum of the system = Initial momentum of the system \\
- [tex]\((m_{\text{astronaut}} \cdot v_{\text{astronaut-final}}) + (m_{\text{gas}} \cdot v_{\text{gas-final}}) = 0\)[/tex] \
4. Rearrange to Solve for the Astronaut's Final Velocity [tex]\((v_{\text{astronaut-final}})\)[/tex]:
- [tex]\((m_{\text{astronaut}} \cdot v_{\text{astronaut-final}}) = -(m_{\text{gas}} \cdot v_{\text{gas-final}})\)[/tex]
- Substituting in the given values: \\
[tex]\(84 \cdot v_{\text{astronaut-final}} = - (0.035 \cdot 875)\)[/tex]
5. Calculate the Astronaut's Final Velocity:
- Solving the equation:
- [tex]\(v_{\text{astronaut-final}} = \frac{-(0.035 \cdot 875)}{84}\)[/tex]
6. Final Result:
- After performing the calculation, the velocity of the astronaut after firing the thruster is approximately [tex]\(-0.365 \, \text{m/s}\)[/tex].
This means the astronaut moves in the opposite direction of the expelled gas with a velocity of about [tex]\(-0.365 \, \text{m/s}\)[/tex] after firing the thruster. The negative sign indicates the direction is opposite to that of the expelled gas.
1. Identify the System and Initial Conditions:
- We have an astronaut in space, initially at rest, with a mass [tex]\( m_{\text{astronaut}} = 84 \)[/tex] kg.
- The astronaut fires a thruster that expels gas with a mass [tex]\( m_{\text{gas}} = 0.035 \)[/tex] kg at a velocity [tex]\( v_{\text{gas}} = 875 \, \text{m/s} \)[/tex].
2. Apply Conservation of Momentum:
- Since there are no external forces, momentum is conserved. \\
- Initially, both the astronaut and the gas are at rest, so the total initial momentum is zero.
3. Write the Momentum Equation:
- Final momentum of the system = Initial momentum of the system \\
- [tex]\((m_{\text{astronaut}} \cdot v_{\text{astronaut-final}}) + (m_{\text{gas}} \cdot v_{\text{gas-final}}) = 0\)[/tex] \
4. Rearrange to Solve for the Astronaut's Final Velocity [tex]\((v_{\text{astronaut-final}})\)[/tex]:
- [tex]\((m_{\text{astronaut}} \cdot v_{\text{astronaut-final}}) = -(m_{\text{gas}} \cdot v_{\text{gas-final}})\)[/tex]
- Substituting in the given values: \\
[tex]\(84 \cdot v_{\text{astronaut-final}} = - (0.035 \cdot 875)\)[/tex]
5. Calculate the Astronaut's Final Velocity:
- Solving the equation:
- [tex]\(v_{\text{astronaut-final}} = \frac{-(0.035 \cdot 875)}{84}\)[/tex]
6. Final Result:
- After performing the calculation, the velocity of the astronaut after firing the thruster is approximately [tex]\(-0.365 \, \text{m/s}\)[/tex].
This means the astronaut moves in the opposite direction of the expelled gas with a velocity of about [tex]\(-0.365 \, \text{m/s}\)[/tex] after firing the thruster. The negative sign indicates the direction is opposite to that of the expelled gas.
