An alternative derivation of ideal rocket’s flight equation (Viva Rocketry! Part 1.3)

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I will derive the ideal rocket’s flight equation differently than what is shown in “Viva Rocketry! Part 1

Let

\Delta m –  the mass of the propellant

m – the mass of the rocket at time t

v – the speed of the rocket and \Delta m at time t

u – the speed of the ejected propellant, relative to the rocket

p_1 – the magnitude of \Delta m‘s momentum

p_2 – the magnitude of rocket’s momentum

For the propellant:

\Delta p_1 = \Delta m \cdot (\boxed {v +\Delta v -u} ) - \Delta m \cdot v

=\Delta m\cdot v + \Delta m\cdot \Delta v -\Delta m\cdot u - \Delta m \cdot v

= -\Delta m \cdot u + \Delta m \cdot \Delta v

where \boxed {v +\Delta v -u} is the speed of \Delta m at t+\Delta t (see “A Thought Experiment on Velocities”)

By Newton’s second law,

F_1 = \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta p_1}{\Delta t}

= \lim\limits_{\Delta t \rightarrow 0} \frac{-\Delta m \cdot u + \Delta m \cdot \Delta v}{\Delta t}.

For the rocket:

\Delta p_2 = (m-\Delta m) \cdot (v +\Delta v) - (m-\Delta m)\cdot v

= (m-\Delta m)(v +\Delta v-v)

= (m-\Delta m)\cdot {\Delta v}

= m \cdot \Delta v - \Delta m \cdot \Delta v

F_2 = \lim\limits_{\Delta t \rightarrow 0} \frac{\Delta p_2}{\Delta t}

= \lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v - \Delta m \cdot \Delta v}{\Delta t}.

By Newton’s third law,

F_2 = -F_1.

Therefore,

\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v - \Delta m \cdot \Delta v}{\Delta t} = - \lim\limits_{\Delta t \rightarrow 0} \frac{-\Delta m \cdot u + \Delta m \cdot \Delta v}{\Delta t}

That is,

\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v - \Delta m \cdot \Delta v}{\Delta t} + \lim\limits_{\Delta t \rightarrow 0} \frac{-\Delta m \cdot u + \Delta m \cdot \Delta v}{\Delta t} = 0.

It implies

\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v - \Delta m \cdot \Delta v -\Delta m \cdot u + \Delta m \cdot \Delta v}{\Delta t} = 0

or,

\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v -\Delta m \cdot u}{\Delta t} = 0.

Since

\lim\limits_{\Delta t \rightarrow 0} \frac{m \cdot \Delta v -\Delta m \cdot u }{\Delta t}= \lim\limits_{\Delta t \rightarrow 0} {\frac{m\cdot \Delta v  -u \cdot \Delta m}{\Delta t}}=m \cdot \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta v}{\Delta t}-u \cdot \lim\limits_{\Delta t \rightarrow 0} \frac{\Delta m}{\Delta t},

\frac{dv}{dt} = \lim\limits_{\Delta t \rightarrow 0}\frac{\Delta v}{\Delta t},

and

\lim\limits_{\Delta t \rightarrow 0}\frac{\Delta m}{\Delta t}= \lim\limits_{\Delta t \rightarrow 0} \frac{m(t) - m(t+\Delta t)}{\Delta t}= -\lim\limits_{\Delta t \rightarrow 0} \frac{m(t+\Delta t) - m(t)}{\Delta t} = -\frac{dm}{dt}

we have

m \cdot \frac{dv}{dt} + u \cdot \frac{dm}{dt} = 0,

the ideal rocket’s flight equation obtained before in “Viva Rocketry! Part 1“.

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One thought on “An alternative derivation of ideal rocket’s flight equation (Viva Rocketry! Part 1.3)

  1. Pingback: When rocket ejects its propellant at a variable rate (Viva Rocketry! Part 1.4) | Vroom

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