Escape velocity and formulas for circular orbits

In summary, the conversation discusses various formulas for circular orbits and how they can also be applied to other types of orbits such as elliptical, parabolic, and hyperbolic. It is pointed out that circular and elliptical orbits have negative energy, while parabolic orbits have zero energy and hyperbolic orbits have positive energy. Furthermore, the formula for circular orbits can also be applied to elliptical orbits by substituting the semi-major axis for the radius.
  • #1
StephenPrivitera
363
0
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?
 
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  • #2
Bound orbits are those with E < 0.

- Warren
 
  • #3
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
 

What is escape velocity and why is it important?

Escape velocity is the minimum speed required for an object to break free from the gravitational pull of a larger body, such as a planet or moon, and continue moving into space. It is important because it determines whether an object will stay in orbit or escape the gravitational pull of a larger body.

What is the formula for calculating escape velocity?

The formula for calculating escape velocity is v = √(2GM/r), where v is the escape velocity, G is the gravitational constant, M is the mass of the larger body, and r is the distance between the two objects.

How is escape velocity related to circular orbits?

In a circular orbit, the velocity of an object is determined by its distance from the center of the larger body and the gravitational force acting on it. If the object's velocity is greater than the escape velocity, it will escape the circular orbit and continue moving into space.

What is the formula for calculating the velocity of an object in a circular orbit?

The formula for calculating the velocity of an object in a circular orbit is v = √(GM/r), where v is the velocity, G is the gravitational constant, M is the mass of the larger body, and r is the distance between the two objects.

How does the mass of the larger body affect escape velocity and circular orbits?

The mass of the larger body directly affects the escape velocity and the velocity of objects in circular orbits. A larger mass will result in a higher escape velocity and a higher velocity for objects in circular orbits, while a smaller mass will result in a lower escape velocity and a lower velocity for objects in circular orbits.

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