## 6.04.2020

### NatureHacker's Rocket Equation

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Momentum initial equals momentum final based on the conservation of momentum.  In the first case we have the rocket and it's fuel moving at initial velocity to the left.  Then the fuel (Me) is ejected and the rocket continues with velocity 2 and the exhaust travels to the right at velocity Ve.

Setting these two situations equal to each-other since momentum is conserved, we can solve for the change in velocity of the rocket as seen in the picture above.

To get acceleration (as seen in the picture below) you divide the right side of the equation by time it took to eject the exhaust.  We realize that mass of exhaust divided by time is simply mass flow rate.

To get force we multiply this by Mass of the rocket (Mr).

So the final equation of force on the rocket is simply:
Force on the rocket = mass flow rate of the exhaust x (Initial forward velocity + Backward Exhaust velocity)

If we consider the rocket with velocity 0 relative to the exhaust, then the equation becomes simply:

Force on Rocket = mdot x Velocity of Exhaust.

Now it is important to realize that this is best case scenario where the exhaust is ejected in a fully contained capsule.  In reality gas flows in all directions while in a vacuum - negating any sort of force on the rocket.  Atmospheric pressure will contain the gas somewhat so the force will be somewhere between zero and what the equation here says.

NASA claims they have an extra term in their rocket equation.  Question them and have them show you proof that they can get their rockets to provide more force on the rocket than the above equation would specify as the maximum possible.

Actually this paper (archive) suggests that maximum thrust is where pressure of the exhaust gas is equal to the ambient pressure.  This would mean that rockets would have zero thrust in a vacuum where ambient pressure is zero and therefore for maximum thrust the pressure of the exhaust would be zero as well... but this seems to contradict the equation they purport to support...

This post (archive) attempts to explain the apparent discrepancy and lends support that high exhaust pressure and low ambient pressure leads to very low thrust.  Thus rockets have no thrust in space.

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