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Say we are launching an unmanned probe (~5000 kg) to some place in the inner solar system.How much fuel could be saved if the rocket (let's say single stage, for simplicity) could be launched from an altitude of 50 km rather than from sea level?
The 50 km change in distance is essentially negligible, but it is by far the most fuel-intensive 50 km of the trip (atmospheric drag, greatest gravitational force from Earth, and this is when the rocket is most massive).
let's say single stage, for simplicity
launched from an altitude
launched from ... jets - but these are not attempting to reach orbital velocity
the gravitational force at the earth's surface is about the same as it is 50km above the surface.
All the fuel being used to put a satellite into orbit goes into kinetic energy because we need the satellite to move quite fast when in orbit. Do yourself a favor and calculate the difference in energy between a change in potential associated with 50km above the earth's surface and the energy of motion of a satellite circling the earth.