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Kinetic energy of a rocket: is this right?

I've looked at the equation for a rocket under no gravitational acceleration and I've learned how to get the equation for its velocity in terms of mass:

\Delta v = u\cdot\ln (\frac{M_0}{M})

, where u is the constant exhaust velocity [relative to the rocket], that all makes sense; then I wondered what the equation for the transfer of kinetic energy would be, and I got all this:

Method:

Force

Spoiler

Velocity in terms of mass [assuming]

Spoiler

Rearranging...

Spoiler

\Delta E_k = u^2 ( M\ln(\frac{M}{M_0}) -\Delta M )

It all seems right, and

E_k

is always greater than 0 for all

0 < M < M_0

; I think that it looks like a pretty sensible and cool curve :tongue:

Also,

M = M_0 \Rightarrow E_k = u^2 (M_0\cdot 0 - 0) = 0

, so that works...

I was planning on trying to rewrite this in terms of energy density [per kg] of the fuel, but I realised that it wouldn't really be that interesting :coma: