Subtopic: Tsiolkovsky Rocket Equation & Mass Ratio
Question Statement:
A single stage chemical rocket, having an initial mass of 10,000 kg and structural mass (burnout mass) of 2000 kg, uses a propellant with an effective exhaust velocity of 2500 m/s. The maximum velocity increment $\Delta V$ (in km/s) of the rocket in gravity-free, vacuum space is _________ (round off to 2 decimal places).
Official Answer Key:
4.02
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.49)**
**1. Intuitive Physical Setup & Given Parameters**:
- Initial mass: $m_0 = 10000\text{ kg}$
- Burnout mass: $m_b = 2000\text{ kg}$
- Exhaust velocity: $c = 2500\text{ m/s}$
**2. Step-by-Step Propulsion Derivation**:
1. **Mass Ratio ($R_m$)**:
$R_m = \frac{m_0}{m_b} = \frac{10000}{2000} = 5.0$
2. **Tsiolkovsky Rocket Equation**:
$\Delta V = c \ln\left( \frac{m_0}{m_b} \right) = 2500 \times \ln(5.0)$
3. **Evaluating Numerical Value**:
$\ln(5.0) \approx 1.609438$
$\Delta V = 2500 \times 1.609438 = 4023.6\text{ m/s} = 4.0236\text{ km/s}$
Rounding to two decimal places:
$\Delta V \approx 4.02\text{ km/s}$
This falls inside the official range **[3.95, 4.05] km/s**.
**3. Comprehensive Options & Distractor Analysis**:
- **Question Type**: Numerical Answer Type (NAT).
- **Official Range**: **3.95 to 4.05 km/s**.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **4.02**
- **Key Formula**: $\Delta V = c \ln(m_0/m_b)$.
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