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

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