Consider $2$ ramjet engines designed to fly at the same altitude and ambient conditions. One of the engines has a convergent$-$divergent nozzle which expands the combustion products to the ambient pressure. The other engine is fitted with a convergent nozzle and operates at choked condition. Considering ideal themodynamic cycles for both the engines, and same flight Mach number $(M_a),$ show that the ratio of the exhaust gas velocities can be expressed as

$\frac{V_{e,CD}}{V_{e,C}}=M_a\sqrt[\phantom{2}]{\frac{(+1)}{2+(-1)M_a^2}}$

where $V_{e,CD}$ is the exhaust velocity of the engine with the convergent$-$divergent nozzle and $V_{e,C}$ is the exhaust velocity of the engine with the convergent nozzle operating at choked condition. $$ is the ratio of specific heats of the working fluid and is assumed to be constant throughout the cycles.

$[$Marks: $15]$