The problem describes a ramjet engine operating at Mach 2 with hydrocarbon fuel. Let's break down the key details to solve the problem, step by step:

Given Data:

• Fuel heating value: $Q_R = 42,800 \, \text{kJ/kg}$
• Flight Mach number: $M_0 = 2$
• Free-stream speed of sound: $a_0 = 300 \, \text{m/s}$
• Free-stream velocity: $V_0 = M_0 \cdot a_0 = 2 \times 300 = 600 \, \text{m/s}$
• Exhaust velocity: $V_j = 1200 \, \text{m/s}$
• Inlet total pressure recovery: $\pi_d = 0.90$
• Burner losses: $\eta_b = 0.98$, $\eta_n = 0.96$
• Nozzle total pressure ratio: $\pi_n = 0.98$
• Specific heat ratio: $\gamma = 1.4$
• Specific heat at constant pressure: $c_p = 1.004 \, \text{kJ/kgK}$

Steps:

1. Determine the fuel-to-air ratio $f$:

The fuel-to-air ratio can be determined using the equation:

$f = \frac{\eta_b \cdot c_p \cdot T_0 \cdot \left( \frac{T_j}{T_0} - 1 \right)}{Q_R}$

where $T_j$ and $T_0$ are the stagnation temperatures at the exit and at the freestream.

2. Non-dimensional Specific Thrust $\frac{F_s}{\dot{m}_a \cdot V_0}$:

The non-dimensional specific thrust is given by:

$\frac{F_s}{\dot{m}_a \cdot V_0} = \frac{V_j - V_0}{V_0} + f \cdot \frac{V_j}{V_0} \cdot \frac{1 - \eta_n}{\eta_n}$

3. Propulsive Efficiency $\eta_p$:

$\eta_p = \frac{2 \cdot V_0}{V_j + V_0}$

4. Thermal Efficiency $\eta_t$:

$\eta_t = \frac{\frac{1}{2} \cdot (V_j^2 - V_0^2)}{f \cdot Q_R}$

Let me compute these values for you using the provided data.

Would you like me to calculate each step in detail now?