A rocket accelerates straight up by ejecting gas downwards. In a small time interval ∆t, it ejects a gas of mass ∆m at a relative speed u. Calculate KE of the entire system at t + ∆t and t and show that the device that ejects gas does work = 
∆m u2 in this time interval (neglect gravity).
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
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