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

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Discussion Overview

The discussion revolves around modeling the rocket equation using the derivative of momentum, particularly in the context of Newton's 3rd Law and the gravitational force acting on a rocket. Participants explore the relationship between the forces acting on the rocket, including the force generated by fuel consumption and the gravitational force from Earth.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant proposes an equation incorporating the gravitational force on the rocket and the derivative of momentum, questioning the correctness of their approach and the relative velocity of the rocket.
  • Another participant suggests that gravity does not need to be accounted for in the 3rd Law, emphasizing the need to relate the momentum change of ejected gases to the force on the rocket.
  • A follow-up question is raised about how to model the effect of gravity alongside the force from gas consumption.
  • Further clarification is sought on the necessity of accounting for gravity, with a reference to historical missions like the Apollo lunar module to illustrate the argument.
  • One participant emphasizes the importance of calculating the thrust force required to overcome gravity based on the rate of change of momentum of the ejected gas.

Areas of Agreement / Disagreement

There is disagreement among participants regarding the necessity of accounting for gravity in the rocket equation. Some participants assert that gravity should not be included in the 3rd Law analysis, while others argue for a separate consideration of gravitational effects.

Contextual Notes

Participants express uncertainty about the correct formulation of the rocket equation, particularly regarding the treatment of gravitational forces and the relative velocity of the rocket. There are also unresolved questions about the implications of mass loss over time.

Who May Find This Useful

This discussion may be useful for individuals interested in rocket dynamics, momentum theory, and the application of Newton's laws in aerospace engineering contexts.

Physyx
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TL;DR
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?
 
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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?
 
Andrew Mason said:
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?
 
Physyx said:
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
 
Last edited:

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