A two-stage rocket has stage 1 structural coefficient $\epsilon_1 = 0.10$ and payload fraction $\lambda_1 = 0.20$. What is the stage 1 mass ratio $R_1 = m_{01}/m_{f1}$? (Round to two decimal places).

Answer Key:

3.33

### Solution: By definition, $m_{01} = m_{p1} + m_{s1} + m_{L1}$. $\epsilon_1 = \frac{m_{s1}}{m_{s1} + m_{p1}}$ and $\lambda_1 = \frac{m_{L1}}{m_{01} - m_{L1}}$. Alternatively, $R_1 = \frac{1 + \lambda_1}{\epsilon_1 + \lambda_1} = \frac{1 + 0.20}{0.10 + 0.20} = \frac{1.20}{0.30} = 4.0$. Wait, standard definition: $R = \frac{m_0}{m_f} = \frac{m_s + m_p + m_L}{m_s + m_L} = \frac{1 + \lambda'}{\epsilon + \lambda'}$ where $\lambda' = m_L / (m_s + m_p)$. If $\lambda' = 0.20$ and $\epsilon = 0.10$, $R = \frac{1 + 0.20}{0.10 + 0.20} = 4.0$. Correct: 4.00 (range 3.95 to 4.05).

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