# Rocket Equation from Newton’s 3rd Law, Force of Gravitation, and dp/dt

I am using the derivative of momentum (dp/dt) with Newton’s 3rd Law with the gravitational force of Earth.

$$F - F_{gravity} = \dfrac{\mathrm{d}p}{\mathrm{d}t} \\ F - \dfrac{G \, m_e \, m_r}{r^2} = v \, \dfrac{\mathrm{d}m}{\mathrm{d}t} + ma\\$$

$$F$$ = Force created by fuel (at time t)

$$G$$ = Gravitational Constant

$$m_e$$ = Mass of Earth

$$m_r$$ = Mass of rocket (at time t)

$$r$$ = Distance between Earth and rocket (at time $$t$$)

$$v$$ = Velocity of rocket relative to Earth (at time $$t$$)

$$dm/dt$$ = Instantaneous rate of change of mass of rocket (at time $$t$$)

$$m$$ = Also mass of rocket (at time $$t$$)

$$a$$ = Instantaneous acceleration of rocket (at time $$t$$, equal to $$dv/dt$$)

Is my equation correct for a standard rocket? Would dm/dt be negative as the rocket is losing mass over time?

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• Welcome to the Astronomy Stack Exchange! This question may fit well on the Space Exploration or Physics exchanges, it would be worth checking those to see if there's anything useful. – antispinwards Oct 24 at 19:09