Speed after a certain amount of time

[TheDuck700]

Hello everyone! Today I was wondering how one can know the speed of a craft orbiting a planet, after a certain amount of time.

For example, if I'm at the periapsis, what would be my speed after 2min? If I'm going at 1500km on a certain orbit, what would be my speed after 5min?

It would be easy if the speed would change linearly while going from the apoapsis to the periapsis or from the periapsis to the apoapsis, but I feel like it doesn't work that way.

Bonus question: How does the speed change in an elliptical orbit?

[bigcalm]

You need Newton's laws of gravitational motion (actually, Kepler's will do, but stick with Newton).  Without any force being applied, your craft will obey the law of conservation of angular momentum.  This means, given the parameters of your orbit, you can calculate how fast you're going at any particular point around the orbit.

[FancyMouse]

4 hours ago, TheDuck700 said:

How does the speed change in an elliptical orbit?

https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Position_as_a_function_of_time

Note the step 2 (Kepler's equation) there. That pretty much means there's no general formula and you have to numerically approximate it. It's going to be (very) tedious if you want to do it by hand.

[Spricigo]

4 hours ago, FancyMouse said:

https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Position_as_a_function_of_time

Note the step 2 (Kepler's equation) there. That pretty much means there's no general formula and you have to numerically approximate it. It's going to be (very) tedious if you want to do it by hand.

Calculus, check 

Trigonometry,  check

Will to endure it,  not check. 

 

[Reusables]

You can intermix vis-viva equation and kepler's second law.

Actually, dealing with the time is hard since nonlinear(?) differential equation gets involved. Instead, you can easily get the speed at specific position using vis-viva.  Just find the length of the semi-major axis and distance from the planet. (Well, planet mass is needed as well)

https://en.m.wikipedia.org/wiki/Vis-viva_equation

https://en.m.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion

Wait, is G constant changed in ksp?

EDIT: I have a program which calculates the position of the craft for specific time. Do you want one?

Edited January 26, 2017 by Reusables
See EDIT

[IncongruousGoat]

6 hours ago, FancyMouse said:

https://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Position_as_a_function_of_time

Note the step 2 (Kepler's equation) there. That pretty much means there's no general formula and you have to numerically approximate it. It's going to be (very) tedious if you want to do it by hand.

I second this sentiment, having actually (due to an excess of zeal and an equal lack of judgement) written a program to do that very numerical approximation. It's really not worth anyone's time (And no, I'm not going to dig up that code. I'd rather re-write the whole thing from sratch than distribute that... thing). If you really want to do the math, Newton's laws of gravitational motion are almost certainly the way to go.

[Reusables]

31 minutes ago, IncongruousGoat said:

I second this sentiment, having actually (due to an excess of zeal and an equal lack of judgement) written a program to do that very numerical approximation. It's really not worth anyone's time (And no, I'm not going to dig up that code. I'd rather re-write the whole thing from sratch than distribute that... thing). If you really want to do the math, Newton's laws of gravitational motion are almost certainly the way to go.

I'm certain that it's worth the time for simple simulation purpose. The approximate solution makes it easy to provide realistic skybox with a bit of perturbation in parameters, like what's done in Stellarium. AFAIK, even apollo program used the keplerian approximation on the SOI.

Also, it's easy to write a program to get the approximate solution. It cost me less than half an hour.

[TheDuck700]

Originally I asked this question to know the speed relative to a target for a rendez-vous. Knowing the position / speed of the target and my own speed / position, I wanted to know what would be our relative speed at the closest approach.I know it's given in KSP now but I wanted to know how it one could be able to calculate it.

The vis-viva equation doesn't help much in that regard since I wanted I already knew the speed and distance to the orbiting body, and neither does the law of conservation of momentum since it would require me to measure the time that the craft takes to travel a certain portion of its orbit and that felt like cheating (Measuring is too easy and only gives an approximation compared to a proper calculus)

Kepler's equation seems to be what I was looking for even if it seems a bit complicated. I won't need a program to calculate it though, I'd prefer making one myself if I need to, it's more fun this way right?

Anyway thanks to everyone for giving me more insight on the subject!