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A rocket's required mass ratio as a function of effective exhaust velocity ratio. The classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity and can thereby move due to the ...
In the relativistic case, the equation is still valid if is the acceleration in the rocket's reference frame and is the rocket's proper time because at velocity 0 the relationship between force and acceleration is the same as in the classical case. Solving this equation for the ratio of initial mass to final mass gives
Konstantin Tsiolkovsky wrote about using rotation to create an artificial gravity in space in 1903. [1] Herman Potočnik introduced a spinning wheel station with a 30-meter diameter in his Problem der Befahrung des Weltraums (The Problem of Space Travel). He even suggested it be placed in a geostationary orbit. [2]
Rocket mass ratios versus final velocity calculated from the rocket equation. The Tsiolkovsky rocket equation, or ideal rocket equation, can be useful for analysis of maneuvers by vehicles using rocket propulsion. [2] A rocket applies acceleration to itself (a thrust) by expelling part of its mass at high speed
Minimizing the mass of propellant required to achieve a given change in velocity is crucial to building effective rockets. The Tsiolkovsky rocket equation shows that for a rocket with a given empty mass and a given amount of propellant, the total change in velocity it can accomplish is proportional to the effective exhaust velocity.
A space vehicle's flight is determined by application of Newton's second law of motion: =, where F is the vector sum of all forces exerted on the vehicle, m is its current mass, and a is the acceleration vector, the instantaneous rate of change of velocity (v), which in turn is the instantaneous rate of change of displacement.
This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the x-axis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the x-axis. In that case θ = 0, and cos θ = 1, which gives:
This equation can be rewritten in the following equivalent form: = / The fraction on the left-hand side of this equation is the rocket's mass ratio by definition. This equation indicates that a Δv of n {\displaystyle n} times the exhaust velocity requires a mass ratio of e n {\displaystyle e^{n}} .