$(8$ pts$.)$ It was shown in class that the thermal efficiency of an ideal turbojet can be expressed as ${}_{th}=1-\frac{1}{{}_r{}_c}.$ Verify this relation using a different approach by starting the derivation using the "kinetic energy" definition of thermal efficiency, i$.$e$.,{}_{th}=\frac{\frac{1}{2}(v_o^2-v_0^2)}{Q_{tn}},$ where $Q_{in}$ would be the head added per unit mass, and introduce the appropriate enthalpies. This problem will help clarify the role of kinetic energy in the thermal efficiency of the gas turbine cycle.