A rocket takes off from rest and leaves the burnt gases with speed u w.r.t. rocket. Find the maximum velocity of rocket if 50% of the total mass of rocket is fuel.

Solution

Let m₀ be the total mass of rocket.
The velocity of the rocket when m gram of fuel gets burnt is,

straight v space equals space straight u space ln space open parentheses fraction numerator straight m subscript straight o over denominator straight m subscript straight o space minus space straight m end fraction close parentheses

Maximum velocity is acquired by the rocket when the whole fuel will burn out.
Since mass of the fuel is 50% of m₀, thus

M a x i m u m space v e l o c i t y comma space v subscript m a x end subscript space equals space u space ln space 2 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space u space cross times space 0.623

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