How much velocity can a canister of fuel give a spaceship?

Question

I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of the ship $M$ and the amount of energy in the fuel $E$. So the energy density of the fuel is, $\frac{E}{m} = p$, which is constant. So the amount of velocity gained by an increment of fuel is equal to $\sqrt{\frac{p\cdot{dm}}{M+m}} = dv$. Since $dE = p\cdot{dm} = \frac{1}{2}(m+M)dv^2$. In order to find $v$, I needed to integrate the left hand side from m equals the total mass of the fuel to $m=0$ (assuming the canister is massless).

Is this correct? How, (if at all) is it possible to evaluate this integral?

P.S. I think I'm way off from this derivation https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

Can someone please lead me in the right direction?

Answer 1

${d\over dv}v^2=2v$ Hence $dv^2=2vdv$ Therefore your equation would be Integral of $\int {p\cdot dm\over m+M}= \int vdv$

Answer 2

The equation in the mentioned Wikipedia page is a special case of the general problem of motion equation for a system with variable mass that comes from the conservation of the momentum of the system. You can see the derivation of the pertinent equation at this page.

The pure energetic approach in OP is not correct because the energy is conserved only in the full system, given by the rocket and the combusted fuels and also the heat produced in the combustion. The key parameter that determines the velocity of the roket is the expulsion velocity of the ejected propellant, that depends on the constructive characteristics of the roket.