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A man paints with his brains and not with his hands. ..."Michelangelo (1475-1564, Italian sculptor, painter, architect and poet)"

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 Author Topic: rocket equation  (Read 11598 times) 0 Members and 1 Guest are viewing this topic. Click to toggle author information(expand message area).
Fu-Kwun Hwang
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 « Embed this message on: January 09, 2010, 10:55:06 am »

A typical rocket engine can handle a significant fraction of its own mass in propellant each second, with the propellant leaving the nozzle at several kilometres per second. This means that the thrust-to-weight ratio of a rocket engine, and often the entire vehicle can be very high, in extreme cases over 100.
It can be shown that the net thrust of a rocket is:

$F_n = \dot{m}\;v_{e}$

where:

$\dot{m} =\,$propellant flow (kg/s or lb/s)

$v_{e} =\,$the effective exhaust velocity (m/s or ft/s)

The $v_{e}$ of a rocket engine is often almost constant in a vacuum, but in practice the effective exhaust velocity of rocket engines goes down when operated within an atmosphere as the atmospheric pressure goes up. In space, the effective exhaust velocity is equal to the actual exhaust velocity. In the atmosphere, the two velocities are close in value.

The Tsiolkovsky rocket equation, or ideal rocket equation, is a mathematical equation that relates the delta v with effective exhaust velocity and the initial and end mass of a rocket.

The equation is named after Konstantin Eduardovich Tsiolkovskii|Konstantin Tsiolkovsky who independently derived it and published it in his 1903 work(К. Э. Циолковский, Исследование мировых пространств реактивными приборами, 1903. It is available online http://epizodsspace.airbase.ru/bibl/dorev-knigi/ciolkovskiy/sm.rar here in a RARed PDF). 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.

For any such maneuver (or journey involving a number of such maneuvers):

:$\Delta v\ = v_\text{e} \ln \frac {m_0} {m_1}= v_\text{e} \ln \frac {m_0} {m_0-\dot{m}t}$

where:
:$m_0$ is the initial total mass, including propellant.
:$m_1$ is the final total mass.
:$v_\text{e}$ is the effective exhaust velocity. ($v_\text{e} = I_\text{sp} \cdot g_0$)
:$\Delta v\$ is delta-v.

Units used for mass or velocity do not matter as long as they are consistent.

The following is a simulation of a rocket.
You can change
1. ratio: The initial rocket mass/ final rocket mass ratio,
2. pratio: propellant mass/ total mass
3. dmdt: $\dot{m}$
4. T: total acceleation time
5. u: $v_{e}$
It will draw it's displacement as a function of time x(t) and velocity as a function of time v(t).

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A man paints with his brains and not with his hands. ..."Michelangelo (1475-1564, Italian sculptor, painter, architect and poet)"