Description
Rocket engines produce the same force regardless of their velocity. A rocket acting on a fixed object, as in a static firing, does no useful work at all; the rocket's stored energy is entirely expended on accelerating its propellant to hypersonic speed. But when the rocket moves, its thrust acts through the distance it moves. Force acting through a distance is the definition of mechanical energy or work. So the farther the rocket and payload move during the burn, (i.e. the faster they move), the greater the kinetic energy imparted to the rocket and its payload and the less to its exhaust.
This can be easily shown. The mechanical work can be defined as
where is the kinetic energy, is the force (the thrust of the rocket which is considered constant), and is the distance. Differentiating with respect to time, we obtain
or
where is the velocity. Dividing by the instantaneous mass to express this in terms of specific energy, we get
where is the acceleration vector.
Thus it can be readily seen that the rate of gain of specific energy of every part of the rocket is proportional to speed, and given this the equation can be integrated to calculate the overall increase in specific energy of the rocket.
However, integrating this is often unnecessary if the burn duration is short. For example as a vehicle falls towards periapsis in any orbit (closed or escape orbits) the velocity relative to the central body increases. Briefly burning the engine (an "impulsive burn") prograde at periapsis increases the velocity by the same increment as at any other time . However, since the vehicle's kinetic energy is related to the square of its velocity, this increase in velocity has a disproportionate effect on the vehicle's kinetic energy; leaving it with higher energy than if the burn were achieved at any other time.
It may seem that the rocket is getting energy for free, which would violate conservation of energy. However, any gain to the rocket's energy is balanced by an equal decrease in the energy the exhaust is left with. When expended lower in the gravitational field, even if the exhaust is left with more kinetic energy, it is left with less total energy. The effect would be even stronger if the exhaust speed could be made equal to the speed of the rocket, then the exhaust would be left without kinetic energy, so the total energy of the exhaust would be as low as its potential energy. Contrast this to the situation of static firing: as the speed of the engine is zero its specific energy does not increase at all, with all chemical energy of the fuel being converted to the exhaust's kinetic energy.
At very high speed the mechanical power imparted to the rocket can even exceed the total power liberated in the combustion of the propellants, and this may also seem to violate conservation of energy. But the propellants in a fast moving rocket carry energy not only chemically but also in their own kinetic energy, which at speeds above a few km/s actually exceed the chemical component. When these propellants are burned, some of this kinetic energy is transferred to the rocket along with the chemical energy released by burning. This can make up for what seems like an extremely low efficiency early in the rocket's flight when it is moving only slowly. Most of the work done by a rocket early in flight is "invested" in the kinetic energy of the propellant not yet burned, part of which they will release later when they are burned.
Read more about this topic: Oberth Effect
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