I want to try to derive the rocket equation and then add additional effects like gravity, air resistance, etc. Here's the equation that I found online:
p (momentum) = mass * velocity
F (force) = Δp / Δt
The Attempt at a Solution
pi = mv
Here is where the problem is. I do this:
pfinal = (m - dm)(v + dv) - dm(v - ve)
But online solutions show that it is either (m + dm) OR + dm(v - ve), not minus.
Anyways, continuing to solve with my equation gets:
pi = pfinal (conservation of momentum)
mv = (m - dm)(v + dv) - dm(v - ve)
mv = mv + mdv - vdm - dmdv - vdm + vedm
Again, here the vdm should cancel, but they dont't.
mdv - dmdv = 2*vdm - vedm
Here, I read that dmdv is considered a higher order term so it is negligible.
mdv = dm(2v - ve)
So I got to this point but the equation is wrong. It is obvious that the mass of the rocket time dv (i.e. its change in momentum) doesn't equal the mass of the exhaust times 2v - ve, it should just be v - ve. But even that is wrong since it is just dm times ve. So my question is what am I missing?