A launch vehicle operating in gravity-free vacuum has an effective exhaust velocity $c = 3200\text{ m/s}$. The initial launch mass is $m_0 = 10000\text{ kg}$ and the final burnout mass after propellant consumption is $m_f = 2000\text{ kg}$. What is the total ideal velocity increment $\Delta v$ (in m/s) imparted to the vehicle?

Answer Key:

5149.7

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Mass Ratio:** $m_0 / m_f = 10000 / 2000 = 5$. 2. **Stage 2: Logarithm Evaluation:** $\ln(5) \approx 1.6094379$. 3. **Stage 3: Velocity Increment:** $\Delta v = c \ln(m_0/m_f) = 3200 \times 1.6094379 = 5150.2\text{ m/s}$. 4. **Stage 4: Range:** 5145 to 5155 m/s.

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