A rocket accelera

At time t,

Kinetic energy

Now at time t+Δt,

Rocket loses mass Δm and gains velocity Δv and releases gas with relative velocity of u

Velocity of rocket w.r.t ground

Velocity of gas w.r.t ground

∴ total kinetic energy of the system w.r.t ground

[neglecting terms with as both of them are very small]

Kinetic gained in Δt interval,

----(1)

∵ there is no external force ∴ internal forces must cancel out,

Putting the above in equation (1), we get,

By work energy theorem,

Work done = change in kinetic energy

W

∴ the device that ejects gas does work = ∆m u² in this time interval