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A method of determining the optimal thrust routine for a rocket engine to produce maximum altitudes for a two-stage launch vehicle. Assuming a flat earth and constant gravitational field, constant and exponential atmospheric densities are considered. A step-by-step method of adding complexity is used to go from a single-stage to the two-stage rocket. At staging, the mass, engine characteristics, and drag profile of the booster all undergo changes, and the switching function can jump from negative to positive values without passing through zero. Intermediate coasting arcs, intermediate variable-thrust arcs, and up to five control switches are all unique to the two-stage trajectory. Although these differences would be lessened by the inclusion of realistic atmospheric drag profiles, they will probably exist if the liftoff acceleration of the sounding rocket is high. A numerical example indicates the improvement possible by optimizing the thrust history. Most real trajectories will never reach the switching plane, so optimal trajectories in most cases utilize full thrust throughout.

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