Escape velocity and formulas for circular orbits

[StephenPrivitera]

In my class, we developed a list of formulas for circular orbits. One of them is E/m=1/2v²-GM/r=constant. To derive escape velocity we find for what v does E=0. But an orbit of this nature is certainly not circular! How can we apply the formula?

 

[chroot]

Bound orbits are those with E < 0.

- Warren

 

[Janus]

The formula given is true for all orbits; whether they be circular, eliptical, parabolic or hyperbolic.

As Warren pointed out, circular or eliptical orbits will have E<0

A parabolic orbit (the one followed by an object traveling exactly at escape velocity) E=0

For E > 0, you get a hyperbolic orbit.

One thing about circular orbits:

v= [squ](GM/r) at all points. if you substitute this for v in the formula you have it reduces to

E=-GMm/2r

Now, this turns out to be also true for eliptcal orbits if you substitute the semi-major axis(a) for r. (the semi-major axis is half of the longest dimension of the ellipse. It is also the Average length of the radius vector over the course of an orbit. )

This gives

E = -GMm/2a