Greetings all,
So i've been an orbinaut for a few years now but i've never really understood the math and physics behind some of the manuevers involved in spaceflight so I figured what a better use of my day than to try and learn. So I began at the beginning and wanted to learn the math behind launching a rocket.
Im using Orbiter 2k10 and the Model Rockets addon using the 4mRocket for those that are familiar with it although i've changed some of the variables such as the masses, thrust and burntime in the config file.
I"m going to go into detail about my learning process and point out some areas i know i'm making mistakes in.
First, my reference are the following links:
http://www.rocketmime.com/rockets/rckt_eqn.html
http://www.rocketmime.com/rockets/qref.html
and naturally http://www.wikipedia.org
Starting at the beginning here i'll list some key values:
Empty Mass (Me) = 4.8
Fuel Mass (Mf) = 10.0
Rocket Mass (M) = 14.8
Rocket Area (A) = 0.2
Thrust (T) = 800
BurnTime (t) = 26
Impulse (I) = 20,800
Gravitational Accel (g) = 9.8
Wind Resistance (k) = 0.5*rho*drag*area
k = 0.5 * 1.22 * 0.75 * 0.2
k = 0.0915
***One thing that should be noted here is that 0.75 for the rocket's drag, however this is a estimation of the avg rocket according to my reference. I do not have the formula for computing the drag of the rocket given the parameters listed here.
Gravitational Force (Gf)= M * g
Gf = 4.8 * 9.8
Gf = 47.04
Then the author of the website suggested to formulas to help simplify future math, those formulas will be called 'q' and 'x' for this example:
Q Value (q) = sqrt( (T - mg) / m )
q = sqrt( (800 - 4.8 * 9.8) / 4.8 )
q = sqrt( 752.96 / 4.8 )
q = sqrt( 156.8667 )
q = 12.5246
X Value (x) = 2kq / m
x = (2 * 0.0915 * 12.5246) / 4.8
x = 2.2920018 / 4.8
x = 0.4775
Now we have enough information to start with the three equations for Velocity at MECO, Altitude at MECO, and Altitude at Apogee:
Vmeco = q * [( 1 - e^-xt ) / ( 1 + e^-xt )]
Vmeco = 12.5246 * [( 1 - e^-0.4775 * 26 ) / ( 1 + e^-0.4775 * 26 )]
Vmeco = 12.5246 * [( 1 - e^-12.415) / ( 1 + e^-12.415)]
Vmeco = 12.5246 * [(1 - 0.0000004057) / ( 1 + 0.0000004057)]
Vmeco = 12.5246 * 0.9999995943 / 1.0000004057
Vmeco = 12.5246
so what this tells me is that the (1 - e^-xt) and the (1 + e^-xt) cancel each other and q = Vmeco... which clearly isn't right, and since I can't go any farther I have to stop here. I know im making a mistake in my math somewhere up the line which is why I was so detailed in my post.
Im going to include the other two formuals as best I understand them here also for checking:
Altmeco = (-m/2k) * [ ln( (T - mg - kv^2) / (T - mg) ) ]
Altcoast = (m/2k) * [ ln( (mg + kv^2) / mg ) ]
Altaposis = Altmeco + Altcoast
so thats where I am. Any input anyone has on my math and equations or any better, easier to understand resources would be much appreciated. Thanks all.
So i've been an orbinaut for a few years now but i've never really understood the math and physics behind some of the manuevers involved in spaceflight so I figured what a better use of my day than to try and learn. So I began at the beginning and wanted to learn the math behind launching a rocket.
Im using Orbiter 2k10 and the Model Rockets addon using the 4mRocket for those that are familiar with it although i've changed some of the variables such as the masses, thrust and burntime in the config file.
I"m going to go into detail about my learning process and point out some areas i know i'm making mistakes in.
First, my reference are the following links:
http://www.rocketmime.com/rockets/rckt_eqn.html
http://www.rocketmime.com/rockets/qref.html
and naturally http://www.wikipedia.org
Starting at the beginning here i'll list some key values:
Empty Mass (Me) = 4.8
Fuel Mass (Mf) = 10.0
Rocket Mass (M) = 14.8
Rocket Area (A) = 0.2
Thrust (T) = 800
BurnTime (t) = 26
Impulse (I) = 20,800
Gravitational Accel (g) = 9.8
Wind Resistance (k) = 0.5*rho*drag*area
k = 0.5 * 1.22 * 0.75 * 0.2
k = 0.0915
***One thing that should be noted here is that 0.75 for the rocket's drag, however this is a estimation of the avg rocket according to my reference. I do not have the formula for computing the drag of the rocket given the parameters listed here.
Gravitational Force (Gf)= M * g
Gf = 4.8 * 9.8
Gf = 47.04
Then the author of the website suggested to formulas to help simplify future math, those formulas will be called 'q' and 'x' for this example:
Q Value (q) = sqrt( (T - mg) / m )
q = sqrt( (800 - 4.8 * 9.8) / 4.8 )
q = sqrt( 752.96 / 4.8 )
q = sqrt( 156.8667 )
q = 12.5246
X Value (x) = 2kq / m
x = (2 * 0.0915 * 12.5246) / 4.8
x = 2.2920018 / 4.8
x = 0.4775
Now we have enough information to start with the three equations for Velocity at MECO, Altitude at MECO, and Altitude at Apogee:
Vmeco = q * [( 1 - e^-xt ) / ( 1 + e^-xt )]
Vmeco = 12.5246 * [( 1 - e^-0.4775 * 26 ) / ( 1 + e^-0.4775 * 26 )]
Vmeco = 12.5246 * [( 1 - e^-12.415) / ( 1 + e^-12.415)]
Vmeco = 12.5246 * [(1 - 0.0000004057) / ( 1 + 0.0000004057)]
Vmeco = 12.5246 * 0.9999995943 / 1.0000004057
Vmeco = 12.5246
so what this tells me is that the (1 - e^-xt) and the (1 + e^-xt) cancel each other and q = Vmeco... which clearly isn't right, and since I can't go any farther I have to stop here. I know im making a mistake in my math somewhere up the line which is why I was so detailed in my post.
Im going to include the other two formuals as best I understand them here also for checking:
Altmeco = (-m/2k) * [ ln( (T - mg - kv^2) / (T - mg) ) ]
Altcoast = (m/2k) * [ ln( (mg + kv^2) / mg ) ]
Altaposis = Altmeco + Altcoast
so thats where I am. Any input anyone has on my math and equations or any better, easier to understand resources would be much appreciated. Thanks all.
