Question
Derive the expression for the velocity of rocket at any instant in the case of rocket propulsion in the presence of gravity.
Solution
Consider, a rocket of mass m0 take off with velocity v0 from ground. 
Let the fuel burn at the rate of dm/dt and burnt gases eject with velocity u w.r.t. rocket.
Let at any instant t, v be the velocity of rocket and m be the mass of rocket.
Therefore velocity of burnt gas ejected w.r.t. ground is (v – u).
Here, fuel is burning at the rate of dm/dt.
Therefore in time dt, dm mass of the fuel will burn and velocity of rocket increases by dv.
Here, we have
'm' is the mass of rocket, and
dm is the mass of gas ejected.
As the fuel burns the mass of rocket decreases.
Therefore, to calculate the velocity of rocket at any instant in terms of mass of rocket, m must be replaced by ‘dm’. 
