Question Statement:
An ideal rocket has characteristic exhaust velocity of 1200 m/s, mass flow rate of
75 kg/s, thrust coefficient of 1.5, and nozzle throat area of 0.025 m^2. The chamber
pressure in kPa and the specific impulse due to gravity in seconds are ________,
respectively. Assume that the acceleration due to gravity is 9.8 m/s^2.
Options:
-
(A) 3600 and 183.67
-
(B) 4600 and 190.51
-
(C) 3600 and 175.23
-
(D) 3500 and 183.67
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2026 Q.39)**
**1. Intuitive Physical Setup & Given Parameters**:
For an ideal rocket engine:
- Characteristic exhaust velocity: $c^* = 1200\text{ m/s}$
- Propellant mass flow rate: $\dot{m} = 75\text{ kg/s}$
- Thrust coefficient: $C_F = 1.5$
- Throat area: $A_t = 0.025\text{ m}^2$
- Gravitational acceleration: $g_0 = 9.8\text{ m/s}^2$
We need to find the chamber pressure $p_c$ in $\text{kPa}$ and specific impulse $I_{\text{sp}}$ in seconds.
**2. Step-by-Step Rocket Propulsion Derivation**:
1. **Chamber Pressure ($p_c$) Calculation**:
The characteristic velocity $c^*$ is defined by:
c^* = \frac{p_c A_t}{\dot{m}}
Solving for chamber pressure $p_c$:
p_c = \frac{\dot{m} c^*}{A_t} = \frac{75 \times 1200}{0.025} = \frac{90000}{0.025} = 3,600,000\text{ Pa} = 3600\text{ kPa}
2. **Thrust ($F$) and Specific Impulse ($I_{\text{sp}}$) Calculation**:
Thrust is related to chamber pressure and throat area by the thrust coefficient:
F = C_F p_c A_t = 1.5 \times (3.6 \times 10^6\text{ Pa}) \times 0.025\text{ m}^2 = 135,000\text{ N}
Specific impulse with respect to standard gravity is:
I_{\text{sp}} = \frac{F}{\dot{m} g_0} = \frac{135000}{75 \times 9.8} = \frac{1800}{9.8} \approx 183.673\text{ s}
Alternatively, using effective exhaust velocity $c = C_F c^*$:
c = 1.5 \times 1200 = 1800\text{ m/s}
I_{\text{sp}} = \frac{c}{g_0} = \frac{1800}{9.8} \approx 183.67\text{ s}
3. Combining results: $p_c = 3600\text{ kPa}$ and $I_{\text{sp}} = 183.67\text{ s}$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A) 3600 and 183.67**: ✅ **CORRECT**. Both chamber pressure ($3600\text{ kPa}$) and specific impulse ($183.67\text{ s}$) are exact.
- **(B) 4600 and 190.51**: ❌ **INCORRECT**. Incorrect chamber pressure calculation.
- **(C) 3600 and 175.23**: ❌ **INCORRECT**. Divides by incorrect gravity value or omits $C_F$.
- **(D) 3500 and 183.67**: ❌ **INCORRECT**. Slight arithmetic error in chamber pressure.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **A**
- **Exam Takeaway**: Master the fundamental rocket relations: $p_c = \frac{\dot{m}c^*}{A_t}$, $c = C_F c^*$, and $I_{\text{sp}} = \frac{c}{g_0}$.