A rocket accelerates straight up by ejecting

Question:

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 = (1/2) ∆m u2 in this time interval.

Solution:

M is the mass of the rocket at any time t

v is the velocity of the rocket

∆m is the mass of the gas that is ejected during the time interval ∆t

Therefore, K = 1/2 u2∆m

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