# Elliptical orbit

i was wondering, is there a particular formula to calculate the velocity of a object in an elliptical orbit. Lets say a satellite orbiting around the earth, and the orbit is elliptical, so how do u calculate the velocity at a certain distance from earth. I tried using the v^2=GM/r, but thats only for circular orbits.
thx for ur time

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enigma
Staff Emeritus
Gold Member
Welcome to the forums!

The generalized form is called the Vis-Viva equation:

$$V=\sqrt{\mu*(\frac{2}{r}-\frac{1}{a})}$$

Where $\mu$ is G*M or 398600.4 km^3/sec^2 for Earth,
r is the distance from the center of the Earth and
a is the semimajor axis of the ellipse.

You'll see that for a circular orbit, a = r for all points on the "ellipse" and you get the expected $$\sqrt{\frac{\mu}{r}}$$. You can also get the escape velocity by plugging in infinity for a.

There are two things that must be remembered

1. Conservation of Angular Momentum
2. Conservation of Energy at any moment

Writing the above equations as function of r,v
and calulate r or v whatever required

THanx alot for the help, even though in high school we haven't learned that formula yet, but it was really helpfull.

enigma
Staff Emeritus
Gold Member
Originally posted by matrix_204
even though in high school we haven't learned that formula yet,
Do you understand it? The way you worded that, it sounds like you didn't.

It really isn't any more difficult than sqrt(mu/r). a is half the distance of the longest line in the ellipse, r is the current position. Plug and chug.

i also found it using the conservation of energy, except with the formula i was a little confused but somehow i got the answer, with it, so i guess that's an alternate way of doing it as well. but the idea of conservation energy was good because thats how much we are taught so far. and i did understood too, n e ways. thnx again

Yes those two equations are basic foundation for deriving formula

General Math or Physics

Each planet moves around the sun in an elliptical orbit. the orbital period, T,of a planet is the timeit takes the planet to go once around the sun. the orbital period of a planet is proportional to the 3/2 power of the length of its semi-major axis. what is the orbial period (in days) of Mercury whose semi-major axis is 58 million km? what is the period (in years) of Pluto whose semi-major axis is 6,000 million km? the semi-major axis of the Eart is 150 million km.
how do you solve this