Rocket velocity including Universal gravitation

[azaharak]

First off this is not homework or coursework, just general interest.


I've been looking to derive the rocket equation which includes the effects of Universal Gravitation. I've been able to derive it assuming near Earth gravity where g is taking as constant acceleration.


d(vrocket)=-[G(Mearth)/(r^2)]dt -(dm/m)(vexaust)

where vexaust is the exhaust speed of the fuel.

My thoughts are to relate dt to (dt/dr)dr which is equal to (1/(vrocket))dr

This gives

d(vrocket)=-[G(Mrocket)(Mearth)/(r^2)]*(1/(vrocket))dr -(dm/m)(vexaust)


If I multiply by (vrocket) throughout this would be good for integration on the left side however it would proprogate to the differential mass term. Since vrocket isn't constant I can not simply integrate the right term (see below)

(vrocket)d(vrocket)=-[G(Mrocket)(Mearth)/(r^2)]dr -(dm/m)(vexaust)(vrocket).


Any ideas, is this even solvable analytically?


Best

AZ

 

[azaharak]

I'm thinking it must be solved numerically?

Anyone?

 

[D H]

http://books.google.com/books?id=SuPQmbqyrFAC&pg=PA128#v=onepage&q&f=false

 

[azaharak]

Thanks for the useful information/read but that doesn't assume universal gravitation, it assumes g=9.8m/s2


It does modulate the impulse delivered by the force of gravity by sin(theta) if the rocket is not traveling straight up.

Thanks, I think the equations need to be solved numerically (there is no analyitcal form).

Best

AZ

 

[D H]

Thanks for the useful information/read but that doesn't assume universal gravitation, it assumes g=9.8m/s2

It specifically does not assume that. Read the text: g is "the gravitational acceleration at the rocket's location".