Tsiolkovsky rocket equation

Tsiolkovsky's rocket equation, named after Konstantin Tsiolkovsky who first derived it, considers the principle of a rocket: a device that can apply an acceleration to itself (a thrust) by expelling part of its mass with high speed in the opposite direction, due to the conservation of momentum.

Related Topics:
Konstantin Tsiolkovsky - Rocket - Thrust - Momentum

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It says that for any maneuver or any journey involving a number of maneuvers:

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:Delta v = v_e ln rac {m_0} {m_1}

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or equivalently

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:m_1=m_0 e^{-Delta v / v_e}      or      m_0=m_1 e^{Delta v / v_e}

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where m_0 is the initial total mass, and m_1 the final total mass and v_e the velocity of the rocket exhaust with respect to the rocket (the specific impulse, or, if measured in time, that multiplied by gravity-on-Earth acceleration).

Related Topics:
Specific impulse - Gravity

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:1-rac {m_1} {m_0}=1-e^{-Delta v / v_e}is the mass fraction (the part of the initial total mass that is spent as reaction mass).

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Delta v (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 negligible, 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.

Related Topics:
Delta v - Elliptic orbit

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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.

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The equation is obtained by integrating the conservation of momentum equation

Related Topics:
Integrating - Conservation of momentum

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:mdv = v_e dm

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for a simple rocket that emits mass at a constant velocity (dm is here the reaction mass; if it is the change of the rocket mass then there is a minus sign in the latter equation).

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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.

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Clearly, to achieve a large delta-v, either m_0 must be huge (growing exponentially as delta-v rises), or m_1 must be tiny, or v must be very high, or some combination of all of these.

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In practice, this has been achieved by using very large rockets (increasing m_0), with multiple stages (decreasing m_1), 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.

Related Topics:
Saturn V - Apollo space program - Ion thruster

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The rocket equation shows a kind of "exponential decay" 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-life" is v_e ln 2 pprox 0.693 v_e

Related Topics:
Exponential decay - Half-life

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