Calculating Velocity During Accent

[ElasticRaven]

I\'m trying to calculate the velocity of my rocket at a given time, assuming I ascend straight up with no worries about wobbling and steering issues. Here is what I have so far:

v(t_f) = t_f + ?₀^t_f(F(t)/m(t)) dt

where

F(t) = T(t) + F_g(t) + F_d(t)

T(t) = thrust at time t

F_g(t) = force of gravity at time t ? -gm(t) at low altitudes

g = Kerbin\'s gravitational accel at sea level = 9.8066 m/s

F_d(t) = force of drag at time t = -½ge^-x(t)/5000C_d(t)v²(t) source

x(t) = altitude at time t = ?₀^tv(t)dt

C_d(t) = mass-weighted drag factor at time t

m(t) = mass at time t

If I work all this out, I get v(t_f) as an integral equation involving not only ?v²(t) dt but also ?e^?v(t)dt dt. How does one solve this? My first thought was to take derivatives until the integrals go away, then solve the ODE. But unless I goofed up, the exponential term causes the integral to remain forever.

My rocket is simple: stock chute, stock pod, Z02 main, stock solid booster. So it\'s easy to account for everything except drag (of course!). Any hints would be appreciated.

[ferram4]

You\'re gonna have to solve this numerically, but I would do this:

^dx/_dt = v(t)

^dv/_dt = ^{(T(t) - ½*m(t)*0.0098*e-x(t)/5000}C_d(t)*v²(t) - ^mu*m(t)/_(RK+x(t))²)/_m(t)

where mu = 3530.35 ^km3/_s²

Then solve it using a standard 4th order Runge-Kutta scheme. It\'ll give you velocity and altitude as functions of time.

If you\'re looking for an analytic solution then I think you\'re out of luck.

Edit: forgot mass in the gravity force term.

[closette]

Attached is a PDF of some restricted solutions I found analytically, on the assumption that 'g' and the drag factor are both constant, and that the Thrust/Weight ratio is held constant (different from constant-thrust).

Even with these simplifications, the functions are hard to evaluate (exponential integrals), and it would be better to use a numerical method such as RK4, although if you use a smaller time-step (0.01s) you can use a simple step-wise linear method, as jebbe and I did for the Goddard Problem challenge thread, here: http://kerbalspaceprogram.com/forum/index.php?topic=7161

Kosmo-not implemented something similar just by using an Excel spreadsheet to update the mass, velocity, thrust etc. in each timestep (referenced in the same thread above). We all learned a lot about the atmosphere and optimal ascent strategies in that thread!

For constant thrust (therefore dm/dt = constant), even assuming a constant-density atmosphere I cannot get an analytical solution. If someone is interested enough I can post the C++ code I use, but you have to edit the thrust, mass, and drag factors for each particular rocket, using information from the KSP Parts Wiki.

[ElasticRaven]

Thanks both. This is great stuff.

[Panichio]

My brain nearly exploded.