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It says that for any maneuver or any journey involving a number of maneuvers:
or equivalently
where is the initial total mass, and the final total mass and the velocity of the rocket exhaust with respect to the rocket (the specific impulse).
( delta v) is the integration over time of the magnitude of the acceleration produced by using the rocket engine (not the acceleration due to other sources such as gravity or drag). For the typical case of an acceleration in the direction of the velocity, this is the increase of the speed. In the case of an acceleration in opposite direction (deceleration) it is the decrease of the speed. Note that gravity or drag also change velocity, but they are not part of the quantity delta-v. Hence delta-v is not simply the change in speed or velocity. However, thrust is often applied in short bursts, and during these short periods the other sources of acceleration may be negligable, and the delta-v of one burst may be simply approximated by the speed change. The total delta-v can simply be found by addition, even though between bursts the magnitude and direction of the velocity changes due to gravity, e.g. in an elliptic orbit.Note that, as mentioned, at any time the magnitude of the acceleration contributes to the delta-v, hence always a non-negative value, regardless of whether the rocket is used for acceleration or deceleration. This again demonstrates that delta-v is not simply the change in speed or velocity: the latter may be zero if we first accelerate and than decelerate, but the delta-v accumulates.
The equation is obtained by integrating the conservation of momentum equation
for a simple rocket that emits mass at a constant velocity ( is here the reaction mass; if it is the change of the rocket mass then there is a minus sign in the latter equation).
The equation does not take into account the discarding of rocket stages; these reduce the mass even more, but unfortunately without providing the corresponding momentum (they can not be ejected with the exhaust speed); however, for the resulting payload this is more advantageous than having to subtract the mass of used fuel tanks and engines from the final .
Although an extreme simplification, the rocket equation captures the essentials of rocket flight physics in a single short equation. It happens that delta-v is one of the most important quantities in orbital mechanics, that quantifies how difficult it is to get from one trajectory to another.
Clearly, to achieve a large delta-v, either must be huge ( growing exponentially as delta-v rises), or must be tiny, or must be very high, or some combination of all of these.
In practice, this has been achieved by using very large rockets (increasing ), with multiple stages (decreasing ), and rockets with very high exhaust velocities. The Saturn V rockets used in the Apollo space program and the ion thrusters used in long-distance unmanned probes are good examples of this.
The rocket equation shows a kind of " exponential decayA quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. Symbolically, this can be expressed as the following differential equation, where N is the quantity and is a positive number called the decay consta" of mass, but not as a function of time, but as a function of delta-v produced. The delta-v that is the corresponding " half-lifeThis article describes the scientific meaning. For the computer game, see Half-Life''. For a quantity subject to exponential decay, the half-life is the time required for the quantity to fall to half of its initial value. Quantities subject to exponential" is