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Background
This is for a video game, "Kerbal Space Program"... I'm sure some of you here have heard of it. It's the type of game you guys would be interested in.
I have built a program to design rockets, but I'm not sure what the optimal thrust-to-weight ratio is. That is, how much fuel I should tack on before the cost of getting it into space exceeds the benefit of having more fuel. I've already figured out how to analytically find the optimal TWR given an environment of constant -9.81 m/s^2 gravity, but... gravity changes via the inverse square law.
Question
I need to solve the following differential equation for ## \frac{dx}{dt} ##. I can solve either one of the two RHS components separately, but not both of them at once.
$$
\frac{d^2x}{dt^2} = \frac{-S}{x^2}+\frac{F_T}{m_i-\dot{m}t}
$$
S, F_T, m_i, and m_dot are constants relating to the specifications of the rocket. All of them are positive.
This is for a video game, "Kerbal Space Program"... I'm sure some of you here have heard of it. It's the type of game you guys would be interested in.
I have built a program to design rockets, but I'm not sure what the optimal thrust-to-weight ratio is. That is, how much fuel I should tack on before the cost of getting it into space exceeds the benefit of having more fuel. I've already figured out how to analytically find the optimal TWR given an environment of constant -9.81 m/s^2 gravity, but... gravity changes via the inverse square law.
Question
I need to solve the following differential equation for ## \frac{dx}{dt} ##. I can solve either one of the two RHS components separately, but not both of them at once.
$$
\frac{d^2x}{dt^2} = \frac{-S}{x^2}+\frac{F_T}{m_i-\dot{m}t}
$$
S, F_T, m_i, and m_dot are constants relating to the specifications of the rocket. All of them are positive.