"One could smear a little bit of solid fuel inside the supersonic diverging nozzle (or inject some liquid fuel). The gasses produced by that additional fuel will be driven to supersonic speed by the gasses coming from the inside of the motor and they will be allowed to expand and accelerate inside the diverging nozzle just like the other gasses. (I believe so, which is not a proof.)"
"This implies the whole rocket motor can be made of solid rocket fuel."
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Solid rocket motors