## Problem Restatement Given- Specific impulse rocket \( I_{sp} = 360 \) s - Zero-gravity, vacuum flight - Propellant fraction \( \zeta \) ranges from .80 to .95 **Task**: Calculate and graph the terminal speed (\( \Delta V \)) as a function of \( \zeta \). --- ## Step 1: Rocket Equation We use the **Tsiolkovsky Rocket Equation**: \[ \Delta V = I_{sp} \cdot g_ \cdot \ln\left(\frac{m_}{m_f}\right) \] Where: - \( I_{sp} \) = specific impulse (s) - \( g_ \) = standard gravity (\( 9.81\,\text{m/s}^2 \)) - \( m_ \) = initial mass (vehicle + propellant) - \( m_f \) = final mass (vehicle after burning propellant) --- ## Step 2: Express Mass Ratio in Terms of Propellant Fraction Propellant fraction \( \zeta \): \[ \zeta = \frac{m_ - m_f}{m_} \implies m_f = m_ (1 - \zeta) \] So, \[ \frac{m_}{m_f} = \frac{m_}{m_(1 - \zeta)} = \frac{1}{1 - \zeta} \] --- ## Step 3: Substitute Values Plug in \( I_{sp} = 360\,\text{s} \), \( g_ = 9.81\,\text{m/s}^2 \): \[ \Delta V = 360 \times 9.81 \times \ln\left(\frac{1}{1 - \zeta}\right) \] \[ \Delta V = 3531.6 \times \ln\left(\frac{1}{1 - \zeta}\right) \] \[ \Delta V = 3531.6 \times \left[-\ln(1 - \zeta)\right] \] \[ \Delta V = -3531.6 \times \ln(1 - \zeta) \] --- ## Step 4: Calculate for Given Range Let's compute a few values for \( \zeta \) between .80 and .95: | \( \zeta \) | \( \Delta V = -3531.6 \times \ln(1 - \zeta) \) (m/s) | |:----------:|------------------------------------:| | .80 | \( -3531.6 \times \ln(.20) = 3531.6 \times 1.6094 = 5687 \) m/s | | .85 | \( -3531.6 \times \ln(.15) = 3531.6 \times 1.8971 = 6702 \) m/s | | .90 | \( -3531.6 \times \ln(.10) = 3531.6 \times 2.3026 = 8134 \) m/s | | .95 | \( -3531.6 \times \ln(.05) = 3531.6 \times 2.9957 = 10576 \) m/s | --- ## Step 5: Graph Here is the relationship, which you can plot: \[ \boxed{ \Delta V = -3531.6 \times \ln(1 - \zeta),\quad .80 \leq \zeta \leq .95 } \] ### Table of Results | Propellant Fraction (\( \zeta \)) | Terminal Speed (\( \Delta V \), m/s) | |:----------------:|:-------------------------:| | .80 | 5687 | | .85 | 6702 | | .90 | 8134 | | .95 | 10576 | --- ### Graph (Description) - **X-axis:** Propellant fraction (\( \zeta \)), from .80 to .95 - **Y-axis:** Terminal speed (\( \Delta V \)), from 500 to 11000 m/s - **Shape:** Exponential increase as \( \zeta \) approaches 1 You can plot this in Excel, Python, or any graphing calculator using the above equation. --- ## **Final Answer** \[ \boxed{ \Delta V = -3531.6 \times \ln(1 - \zeta) } \] For \( \zeta \) between .80 and .95, \(\Delta V\) varies from approximately **5687 m/s to 10576 m/s**. The relationship is nonlinear and can be graphed as described above.