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This post is part of a series of older, now mostly-missing posts, in which I shared math/physics concepts I thought were cool. I was 14/15 when I made them (this was 2021), so a lot of them are not very good, but I'm reuploading them here for posterity's sake (plus, I don't post about physics nearly enough on here, for being a main focus of this site (>_>) I have also preserved their formatting as best as I can, even if that formatting isn't the best either.
"The rocket equation, it's a beautiful thing" - Chris Hadfield, in that Masterclass ad everybody's seen
but how do we know it's a true thing?
First of all, to start this derivation, we're going to be taking a much simpler definition of a rocket. In this, a rocket is anything that ejects part of its mass to propel itself forward (for example, fuel.) This works due to Newtons third law, where the forward force on the rocket is a reaction to the backward force of the ejected material.
To get to deriving the rocket equation, we first start with the conservation of momentum (I briefly gave a description of conservation laws in my last post, and most likely will again for covering noether's theorem, which has been called one of the most beautiful theorems in physics but im going on a tangent aaaaa)
We know that before the ejection, we have the total mass of the rocket and it's fuel going at some velocity
| this will be momentum is written as Mv, M for the mass of rocket with fuel and v for forward velocity
Afterwards, when the fuel is released, it is ejected at some velocity, with its mass being removed from the rocket
| this momentum will be written as (M-D)(v+dv) + D(v-u)
| where D is the mass of the fuel missing, dv is the change in velocity, and u is the velocity of the exhaust let out
anyway, we are then left with the equation
| Mv = (M-D)(v+dv) + D(v-u)
Which we can expand to
| Mv = Mv + Mdv - Dv - Ddv + Dv -Du
Which then simplifies to
| Du = Mdv
| D/M = dv/u
| remembering that D is a change in mass (basically a dM), we can think of "integrating" both sides as
| ln(m0/m) = v/u
| where m0 is the total initial mass and m is the mass after propellant is gone
| thus, **v= u*ln(m0/m) ** which is the rocket equation
this equation is important bc it lets u predict the velocity of a rocket once you account for gravity!