Answer :
To solve this problem, we need to determine how much propellant must be expended to achieve a certain change in velocity [tex](\Delta v)[/tex] using the rocket equation, also known as the Tsiolkovsky rocket equation.
The Tsiolkovsky rocket equation is given by:
[tex]\Delta v = I_{sp} \cdot g \cdot \ln \left( \frac{m_0}{m_f} \right)[/tex]
where:
- [tex]\Delta v[/tex] is the change in velocity (109 m/s in this case).
- [tex]I_{sp}[/tex] is the specific impulse of the thruster (143 seconds).
- [tex]g[/tex] is the acceleration due to gravity [tex](9.81\, m/s^2)[/tex].
- [tex]m_0[/tex] is the initial total mass (dry mass plus propellant).
- [tex]m_f[/tex] is the final total mass (after expelling some propellant).
Let's calculate:
Step 1: Compute [tex]m_0[/tex] using dry mass and initial propellant mass:
[tex]m_0 = 222\, kg + 40\, kg = 262\, kg[/tex]
Step 2: Plug [tex]\Delta v, I_{sp}, g,[/tex] and [tex]m_0[/tex] into the rocket equation and solve for [tex]m_f[/tex]:
[tex]109 = 143 \cdot 9.81 \cdot \ln \left( \frac{262}{m_f} \right)[/tex]
Step 3: Simplify the expression:
[tex]109 = 1402.13 \cdot \ln \left( \frac{262}{m_f} \right)[/tex]
Step 4: Solve for [tex]\ln \left( \frac{262}{m_f} \right)[/tex]:
[tex]\ln \left( \frac{262}{m_f} \right) = \frac{109}{1402.13} \approx 0.0777[/tex]
Step 5: Use exponentiation to solve for [tex]\frac{262}{m_f}[/tex]:
[tex]\frac{262}{m_f} = e^{0.0777} \approx 1.0808[/tex]
Step 6: Solve for [tex]m_f[/tex]:
[tex]m_f = \frac{262}{1.0808} \approx 242.36\, kg[/tex]
Step 7: Calculate the amount of propellant expended:
[tex]\text{Propellant expended} = m_0 - m_f = 262\, kg - 242.36\, kg \approx 19.64\, kg[/tex]
Therefore, the spacecraft must expel approximately 19.60 kg of propellant to achieve a [tex]\Delta v[/tex] of 109 m/s.
The correct multiple-choice answer is: 19.60 kg.
