# How to model a rocket equation from the derivative of momentum?

• Physyx

#### Physyx

TL;DR Summary
Using Newton’s 3rd Law, gravitational force, and derivative of momentum to model a rocket going into space while losing mass.
I am using the derivative of momentum (dp/dt) with Newton’s 3rd Law with the gravitational force of Earth.

F - [Force of gravity on rocket] = dp/dt
F - (G * m_e * m_r / r2 ) = v * dm/dt + 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? Is v relative to the Earth or the expelled gases and why?

Hi Physyx. Welcome to PF!

You do not need to account for gravity in the 3rd law. You just need to relate the rate of change of momentum of the ejected rocket gases to the force on the rocket vehicle and make sure that the force exceeds the force of gravity.

AM

So how would I create a separate equation modeling the effect of gravity on the rocket in addition to the force created by the gas consumption?

You do not need to account for gravity. You just need to relate the rate of change of momentum of the ejected rocket gases to the force on the rocket vehicle.
By this reasoning the Apollo lunar module could happily land and take off from earth. I think you need to be more careful here.
This is detailed in many places...has the OP really looked around?

So how would I create a separate equation modeling the effect of gravity on the rocket in addition to the force created by the gas consumption?
The rocket ejects mass at a certain constant speed. This requires a force provided by the rocket : F = dp/dt . Work out what that force is in terms of the rate of change of momentum of the ejected gas and apply the 3rd law to find the thrust force on the rocket vehicle. What does that force have to be to overcome gravity?

AM

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