Deriving the Equation for Burnout Height in KSP

[bowtiesRcool]

Hi all,

I have been trying to work on an equation I can use to calculate the burnout height of a single stage rocket in vertical flight from the KSP Launchpad where the forces of Drag and Gravity are both variable.

I start with:

SUM(F) = F_thrust + F_drag +F_gravity

where,

F_drag = (0.004892*C_d * P_o)m_rv² e^-h/H

F_gravity = G*Mm_rocket/(r+h)²

F_thrust = some constant

I attempted to modify the derivation of Tsiolkovski's Ideal Rocket Equation to account for the forces of Drag and Gravity where both F_g and F_d are variable.

I have followed the method for deriving the Rocket Equation from MIT's website (here) and ended up with this:

dv = [F_d/m_r - F_g/m_r]dt - (v_edm_r)/m_r

However, the math required to make this equation easily integratabtle is utterly baffling me because of the number of variables present in the function. Does anyone know how to simplify this?

Edited September 12, 2014 by bowtiesRcool
corrected equation (vedmr) should be divided by mr

[LLlAMnYP]

You can take into account that the rockets mass decreases linearly with time. However, that is not entirely correct in KSP, because variable specific impulse with changing air pressure.

[Neil1993]

Tsiolkovsky's famous equation is only really meant for a rocket already in deep space. Where forces like drag and gravity are variable with velocity and altitude, there isn't really any one-step equation that can be used to solve for the altitude. You would have to do a numerical solution using time-steps. I guess you could try to do this with sufficiently steps and then try to curve fit the entire thing, but I don't think this is what you were going for, was it?

[Armchair Rocket Scientist]

Hi all,

I have been trying to work on an equation I can use to calculate the burnout height of a single stage rocket in vertical flight from the KSP Launchpad where the forces of Drag and Gravity are both variable.

I start with:

SUM(F) = F_thrust + F_drag +F_gravity

where,

F_drag = (0.004892*C_d * P_o)m_rv² e^-h/H

F_gravity = G*Mm_rocket/(r+h)²

F_thrust = some constant

I attempted to modify the derivation of Tsiolkovski's Ideal Rocket Equation to account for the forces of Drag and Gravity where both F_g and F_d are variable.

I have followed the method for deriving the Rocket Equation from MIT's website (here) and ended up with this:

dv = [F_d/m_r - F_g/m_r]dt - (v_edm_r)/m_r

However, the math required to make this equation easily integratabtle is utterly baffling me because of the number of variables present in the function. Does anyone know how to simplify this?

Okay, all the force equations look mostly accurate, but thrust is actually a function of height, and the mass of the vehicle is Minitial - t * mass flow rate.

I think you'll get something absurd like d²h/dt² = (thrust as a function of altitude)/(M_initial-propellant mass flow rate * t) - (some constant) * (dh/dt)² * e^-h/H / (M_initial-propellant mass flow rate * t) - G*M_planet/(r+h)².

This is a nightmarish differential equation, including one term with an awkward function of h(t) and t, one term with a nonlinear function of h(t), and one term with an awkward function of h(t) and t multiplied by a nonlinear function of dh/dt. I kind of doubt this is analytically solvable.