GATE AE 2017 • Q49 NAT • 2 Marks Propulsion

Propulsion

Rocket Propulsion

Subtopic: Solid, Liquid & Hybrid Rocket Engines

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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