Assuming that the total momentum of the first stage rocket before and after launch is $p_ 1$ and $p_2$ respectively, ignoring the factors such as gravity and air resistance, it can be obtained according to the law of conservation of momentum: $ P _ 1 = P _ 2 $ $.
During the launch of the first stage rocket, the engine injects an object with a mass of $m$ into the fuel, and accordingly the rocket itself will get a momentum change in the opposite direction $\Delta p = -mv$, where $v$ is the fuel injection speed.
Therefore: $ $ p _ 2 = p _1+\ delta p = p _1-mv $ $.
Since we are concerned about the terminal velocity of the rocket $v_f$, assuming the mass of the rocket is $M$, then: $ $ p _ 1 = mv _ 0, \ quad p _ 2 = mv _ f $.
Substituting the above two equations into the momentum conservation formula and solving $v_f$, the mathematical model of the first stage rocket speed formula can be obtained: $$Mv_0=Mv_f+m v_f$$, $$v_f=\frac{M}{M+m}v_0$$. This formula shows that the terminal velocity of the first stage rocket is related to the initial velocity, rocket quality and fuel quality. In practical application, parameters need to be selected and calculated according to specific conditions.
Applicable conditions of the law of conservation of momentum
1, the system is free from external force or the resultant force is zero;
2. Although the combined external force acting on the system is not zero, when the internal force of the system is much greater than the external force, such as collision and explosion, the momentum of the system can be regarded as approximate conservation.
3. If the system generally does not meet any of the above conditions, the total momentum of the system is not conserved. However, if the system satisfies any of the above conditions in a certain direction, the momentum of the system is conserved in that direction.