Oribtal calculations?

[Rover 6428]

A while ago I had a question that went a little like this:

"Planet Kerbin has a radius of r from its center and gravity of G. Two identical rockets launch from the equator at opposite sides, and start accelerating along the rotation of the planet with the velocity of V. At what time after they finished acceleration did they reach maximum distance from each other, and what was the distance?"

It also said in its prefix that "people attempting this question might want to find out a little about Kepler's laws."

I think I got the answer, but I am not sure if it is right, as I didn't know any good method of doing such a calculation. Could you please help me by telling me how do you find out these variables algebraicly?

I also found the only picture that I had of this question, and here is a graph that went with it

Edited January 10, 2020 by Rover 6428

[Vanamonde]

Seems reasonable to me.  

[UmbralRaptor]

Barring something extremely nice in both how those rockets are accelerating, this will require messing around with numeric methods (eg: using numpy or excel). First to find their trajectories at burnout, and then possibly to deal with the way that Kepler's equation cannot be solved analytically.

[magnemoe]

If the two rockets was launched for opposite sides of the planets and having an equal trajectory they will be most distanced then they reach the new Ap assuming the circulation burn V*T is high enough to raise Ap above the initial Ap.
We need to know the orbital speed they had at initial Ap then they was suborbital too. 
If V*T+ initial speed is lower then needed to raise Ap, initial Ap is the time they are most separated. 

In short you are traveling at V*T+ initial speed at Pe, use this to calculate Ap, twice that is the maximum distance, then calculate then you get there 

[Rover 6428]

3 hours ago, magnemoe said:

In short you are traveling at V*T+ initial speed at Pe, use this to calculate Ap, twice that is the maximum distance, then calculate then you get there 

okay, so algebraecally, how would i find the Ap if I know gravity and the final velocity at Pe? Or how do you calculate an orbital period of such an eliptical orbit?

[magnemoe]

1 minute ago, Rover 6428 said:

okay, so algebraecally, how would i find the Ap if I know gravity and the final velocity at Pe? Or how do you calculate an orbital period of such an eliptical orbit?

I don't know the formulas i just pointed out the logic. 

[UmbralRaptor]

13 hours ago, magnemoe said:

If the two rockets was launched for opposite sides of the planets and having an equal trajectory they will be most distanced then they reach the new Ap assuming the circulation burn V*T is high enough to raise Ap above the initial Ap.
We need to know the orbital speed they had at initial Ap then they was suborbital too. 
If V*T+ initial speed is lower then needed to raise Ap, initial Ap is the time they are most separated. 

In short you are traveling at V*T+ initial speed at Pe, use this to calculate Ap, twice that is the maximum distance, then calculate then you get there 

Pretty much. We need to step through the launch burns (since they are not instantaneous) to get the final orbits. And depending on what the final orbits are, an algebraic solution may not exist.

On 1/9/2020 at 3:21 PM, Rover 6428 said:

 

 

Okay, this helps enormously because of the assumptions that they have the same semi-major axis, orbit orientations such that they are most distant from each-other at apoapsis, both are at periapsis at burnout, and both are at periapsis at the same time. This will require some explanation.

Most fractions of an orbit are a huge hassle to calculate. Full ones are easy, and it takes exactly half an orbit to get between periapsis and apoapsis. But beyond that, you get into the messy realm of converting between mean anomaly (time, sort of), eccentric anomaly, and true anomaly (actual position along an orbit). If the initial orbits are sufficiently different, the maximum distance on the first orbit will have one (or neither!) ships at apoapsis, so will require calculating intermediate distances.

For orbits exactly opposite of each-other with different apoapses/periods, you could still get times given the known start positions (something like 1.5 orbits for one and 2.5 for the other. Exact values depend on the orbits)

The only easy(ish) part is finding some orbital parameters after burnout (given the above assumptions):

Ap + Pe == 2a

Ap = (1+e)*a

Pe = (1-e)*a

Energy = v^2/2 - GM/r = -GM/(2a)

T^2 = (4π^2/GM)*a^3

Where r is current distance, v is current velocity, e is eccentricity, a is semi-major axis, GM is the planet's mass*gravitational_constant, and T is time.

[Zeiss Ikon]

Energy conservation ought to give you the AP (just like it lets you back out bullet velocity from the height reached by a ballistic pendulum).

The kinetic energy the ships have after their boost gets converted back into potential energy.

It'll be a system of equations, though, because you also need to know what the velocity will be at AP to know how much KE stays in that form.  KE_P + PE_P =  KE_A + PE_A.  Remember to calculate your PE from the center of Kerbin (gravitational zero point) rather than from ground level, though, or your height will be off by the PE of Kerbin's radius.  And of course, you have to calculate PE with an inverse square law applied to the value for g.

Might be better to just launch a craft (via hyperdit?), let KSP tell you its Ap height, double that, and subtract Kerbin's diameter.

[tomf]

The vis-viva equation is the one you want to look at. given PE and V you can find a  (the semi major axis) and from pe and a you can get AP