Question

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 \(\text{m}^2\). 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 \(\text{m/s}^2\).

• A. 3600 and 183.67
• B. 4600 and 190.51
• C. 3600 and 175.23
• D. 3500 and 183.67

Hint:

Use \(c^*=p_cA_t/\dot m\) for the chamber pressure, then \(F=C_Fp_cA_t\) and \(I_{sp}=F/(\dot mg)\) for the specific impulse.

Answer 1

The Correct Option is A

Step 1: Combine the two defining formulas symbolically before putting in numbers. The chamber pressure relation is $c^* = p_cA_t/\dot m$, and the thrust relation is $F=C_Fp_cA_t$. Divide the second by $\dot m$: \[ \frac{F}{\dot m} = C_F\left(\frac{p_cA_t}{\dot m}\right) = C_F c^* \] The left side, $F/\dot m$, is exactly the effective exhaust velocity $V_e$. So there is a direct shortcut identity: \[ V_e = C_F\,c^* \]

Step 2: Use this identity to get the effective exhaust velocity straight away. \[ V_e = 1.5 \times 1200 = 1800 \text{ m/s} \] This matches the value found from thrust divided by mass flow, but this route never needed the chamber pressure at all.

Step 3: Get the specific impulse from $V_e$. \[ I_{sp} = \frac{V_e}{g} = \frac{1800}{9.8} \approx 183.67 \text{ s} \]

Step 4: Now find the chamber pressure from the $c^*$ definition on its own. \[ p_c = \frac{c^*\dot m}{A_t} = \frac{1200\times75}{0.025} = 3600000 \text{ Pa} = 3600 \text{ kPa} \] Both quantities agree with the first method, confirming option (A): a chamber pressure of 3600 kPa and a specific impulse of 183.67 s. \[ \boxed{p_c = 3600 \text{ kPa}, \quad I_{sp} = 183.67 \text{ s}} \]