Derivation of rocket propulsion equation - clarification

Question

On a rocket accelerating upward, the forces acting are

• i) the rocket's weight downward

• ii) thrust force upward, due to ejection of gas

We may write $ -u_e dm/dt-mg= F_{ext} $ where $dm/dt$ is the rate of ejection, $u_e$ is the speed of ejected gas w.r.t the rocket and m is the mass of the rocket at some general time instant t=t.

Now my question is, what exactly do we equate this $F_{ext} $ term to, to formulate a differential equation? Two ideas came to my mind.

• i) it seems tempting to set it equal to $m dv/dt$ , as per newton's second law. However it struck me that the more general form of newton's second law is $F_{ext} = dp/dt$ so my second thought was to do

• ii) $F_{ext} = mdv/dt +vdm/dt$

Both of these clearly give different solutions. Which one is correct, and why?

Answer 1

We are interested in the acceleration due to the force, which acts upon the rocket, $$ m a = \sum_i F_i = \frac{dp}{dt} -mg = u_e \frac{dm}{dt} -mg $$ Where I use $\frac{dm}{dt}>0 $. As usually, $a=\frac{dv}{dt} $.