Source Picture C
The Forces Involved: The major forces involved in the rocket launch have been added above where Fg = Force due to gravity= (Ships Mass)* (acceleration of gravity) , Fb = the force that the boosters are generating, and Ff = the frictional force that the air is exerting of the shuttle.
Analysis: We can start very simply by stating Newton's Second Law: ∑F=ma
For our ship we could say: Ms(As)=(Fb-Fg-Ff)
Now lets look at the acceleration of the Shuttle: As=(Fb-Fg-Ff)/Ms
The changing gravitational force: The above equation is good as long as the shuttle is near the surface of the earth which, by the look of things, will not be very long. So we can use some more of Newton's cute little equations:
Fg=(GMsMe)/R^2 •••••> Fg=(GMsMe)/(Re+Hs)^2 •••••> Fg=((6.67*10^-11)MsMe)/(Re+Hs)^2
Me = Mass of the Earth, Re = Radius of the Earth, Hs = Perpendicular Distance Shuttle is from Earth Surface, G = Universal Gravitational Constant.
..........plug it in........... As=(Fb-[((6.67*10^-11)MsMe)/(Re+Hs)^2]-Ff)/Ms
As you can see the force associated with the Earth's gravitational pull will change over time as the shuttle gets further away from the earth's surface.
The changing frictional force: It turns out that the frictional force of air breaks up into more components that change over time and with distance too.
Ff=.5DpAv^2
Of coarse, the density of the fluid that the shuttle is traveling through, air, is decreasing as the shuttle pulls
away from earth. p can be represented as a function of y (height): p=Nve^((-mgy)/KbT)
Nv = Number density given by number of moles per unit of volume of gas, m = the average molecular mass of the fluid, Kb = Boltzmanns constant, T = temperature.
..........plug it in........... Ff=.5D(Nve^((-mgy)/KbT))Av^2
..........plug it in........... As=(Fb-[((6.67*10^-11)MsMe)/(Re+Hs)^2]-[.5D(Nve^((-mgy)/KbT))Av^2])/Ms
don't forget that the frictional air resistance also changes with speed; luckly that is a function of time (just like acceleration).
Interlude...........
Source Picture B
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Now... Back to our regularly scheduled programming
The changing rocket propulsion force: The force that the rocket produces can be broken down in even more
components too: Fb = qVf + (Pn-Pa)Ae
Where q = the mass of fuel ejected per second, Vf = the velocity of the ejected fuel, Pn = the pressure at the nozzle,and Pa = the pressure of the surrounding atmosphere.
Similarly to the density decrease with altitude, (and in fact directly related) the atmosphere pressure changes as a body raises away from earth. Pa = Poe^((-mgy)/KbT)
Everything in the pressure verses altitude equation is the same as the components of the density equation we used to find the changing force of air friction........whew!
..........plug it in........... Fb = qVf + (Pn-(Poe^((-mgy)/KbT)))Ae
..........plug it in...........
As=([qVf + (Pn-(Poe^((-mgy)/KbT)))Ae]-[((6.67*10^-11)MsMe)/(Re+Hs)^2]-[.5D(Nve^((-mgy)/KbT))Av^2])/Ms
The mass of the rocket is also changing over time because the fuel is being burnt away. This is a simple linear relationship and is also a function of time. Ms = Mso - qt
Where Mso = the original mass of the ship, and t = time (probably in seconds)
..........plug it in...........
As=([qVf + (Pn-(Poe^((-mgy)/KbT)))Ae]-[((6.67*10^-11)MsMe)/(Re+Hs)^2]-[.5D(Nve^((-mgy)/KbT))Av^2])/(Mso - qt)
Following the steps that were taken to get to this point, one would see that the only variables left are Distance, Velocity, and Acceleration. The other items left in the equation are all constants. The three remaining variable can all be written in terms of each other so that the acceleration equation could eventually be written in terms of time. For Ben Hoffman's "extra special" derivation of this equation go to CALCULATIONS.