Editing Tsiolkovsky rocket equation (section)

===Most popular derivation===
Consider the following system:

[[File:Tsiolkovsky's Theoretical Rocket Diagram.png|thumb|Tsiolkovsky's theoretical rocket from t = 0 to t = delta_t]]

In the following derivation, "the rocket" is taken to mean "the rocket and all of its unexpended propellant".

[[Newton's second law of motion]] relates external forces (<math>\vec{F}_i</math>) to the change in linear momentum of the whole system (including rocket and exhaust) as follows:
<math display="block">\sum_i \vec{F}_i  = \lim_{\Delta t \to 0} \frac{\vec{P}_{\Delta t} - \vec{P}_0}{\Delta t}</math>
where <math>\vec{P}_0</math> is the momentum of the rocket at time <math>t = 0</math>:
<math display="block">\vec{P}_0 = m \vec{V}</math>
and <math>\vec{P}_{\Delta t}</math> is the momentum of the rocket and exhausted mass at time <math>t = \Delta t</math>:
<math display="block">\vec{P}_{\Delta t} = \left(m - \Delta m \right) \left(\vec{V}  + \Delta \vec{V} \right) + \Delta m \vec{V}_\text{e}</math>
and where, with respect to the observer:
* <math>\vec{V}</math> is the velocity of the rocket at time <math>t = 0</math>
* <math>\vec{V} + \Delta \vec{V}</math> is the velocity of the rocket at time <math>t = \Delta t</math>
* <math>\vec{V}_\text{e}</math> is the velocity of the mass added to the exhaust (and lost by the rocket) during time <math>\Delta t</math>
* <math>m</math> is the mass of the rocket at time <math>t = 0</math>
* <math>\left( m - \Delta m \right)</math> is the mass of the rocket at time <math>t = \Delta t</math>

The velocity of the exhaust <math>\vec{V}_\text{e}</math> in the observer frame is related to the velocity of the exhaust in the rocket frame <math>v_\text{e}</math> by:
<math display="block">\vec {v}_\text{e} = \vec{V}_\text{e} - \vec{V} </math>
thus,
<math display="block">\vec {V}_\text{e} = \vec{V} + \vec{v}_\text{e} </math>
Solving this yields:
<math display="block">\vec{P}_{\Delta t} - \vec{P}_0 = m\Delta \vec{V} + \vec{v}_\text{e} \Delta m - \Delta m \Delta \vec{V}</math>
If <math>\vec{V}</math> and <math>\vec{v}_\text{e}</math> are opposite, <math>\vec{F}_\text{i}</math> have the same direction as <math>\vec{V}</math>, <math>\Delta m \Delta \vec{V}</math> are negligible (since <math>dm \, d\vec{v} \to 0</math>), and using <math>dm = -\Delta m</math> (since ejecting a positive <math>\Delta m</math> results in a decrease in rocket mass in time),
<math display="block">\sum_i F_i = m \frac{dV}{dt} + v_\text{e} \frac{dm}{dt}</math>

If there are no external forces then <math display="inline">\sum_i F_i = 0</math> ([[momentum#Conservation|conservation of linear momentum]]) and
<math display="block">-m\frac{dV}{dt} = v_\text{e}\frac{dm}{dt}</math>

Assuming that <math>v_\text{e}</math> is constant (known as [[Tsiolkovsky's hypothesis]]<ref name="ReactiveFlyingMachines" />), so it is not subject to integration, then the above equation may be integrated as follows:
<math display="block">-\int_{V}^{V + \Delta V} \, dV = {v_e} \int_{m_0}^{m_f} \frac{dm}{m} </math>

This then yields
<math display="block">\Delta V = v_\text{e} \ln \frac{m_0}{m_f}</math>
or equivalently
<math display="block">m_f = m_0 e^{-\Delta V\ / v_\text{e}}</math> or
<math display="block">m_0 = m_f e^{\Delta V / v_\text{e}}</math> or
<math display="block">m_0 - m_f = m_f \left(e^{\Delta V / v_\text{e}} - 1\right)</math>
where <math>m_0</math> is the initial total mass including propellant, <math>m_f</math>  the final mass, and <math>v_\text{e}</math> 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). If <math>v_\text{e}</math> is NOT constant, we might not have rocket equations that are as simple as the above forms. Many rocket dynamics researches were based on the Tsiolkovsky's constant <math>v_\text{e}</math> hypothesis.

The value <math>m_0 - m_f</math> is the total [[working mass]] of propellant expended.

<math>\Delta V</math> ([[delta-v]]) is the integration over time of the magnitude of the acceleration produced by using the rocket engine (what would be the actual acceleration if external forces were absent). In free space, for the case of 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. Of course gravity and drag also accelerate the vehicle, and they can add or subtract to the change in velocity experienced by the vehicle. Hence delta-v may not always be the actual change in speed or velocity of the vehicle.