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Effect of Velocity on Orbital Motion

by Ron Kurtus (revised 19 May 2011)

Although orbital motion of two objects in space can be with respect to the center of mass (CM) between them, it is often more convenient to consider the orbital motion of the smaller object with respect to the larger of the two.

A circular orbit requires a specific tangential velocity at the given separation, with no initial radial velocity to affect the orbit. This situation can occur when two celestial bodies pass near each other at the correct velocities and separation, thus resulting in one orbiting the other. Of more immediate interest is the case of sending up a rocket and aiming it so that its tangential velocity results in a circular orbit.

If the tangential velocity is less than for a circular orbit, the object may go into a small elliptical orbit or even fall into the larger object. If the tangential velocity is greater than for a circular orbit, the object may go into a large elliptical orbit or fly off into space in a parabolic or hyperbolic path.

The equations for a circular orbit and a parabolic path are the starting points in considering the effect of velocity. Also, if the mass of one object is much greater than the other, the equations can be simplified.

Questions you may have include:

This lesson will answer those questions. Useful tool: Units Conversion



Starting points for orbital equations

The path the object follows depends on its tangential velocity. The velocity equations for a circular orbit and for a parabolic path are starting points for examining the relationship between velocity and orbital motion.

Circular orbit

The tangential velocity equation for a circular orbit with respect to the other object is:

vT = √[G(M + m)/R]

where

If one mass is much greater than the other (M >> m), the equation reduces to:

vT = √(GM/R)

Parabolic path

The equation for a parabolic path is:

vT = √[2G(M + m)/R]

Likewise, if M >> m, the equation becomes:

vT = √(2GM/R)

Orbital paths of object

The possible orbital paths of an object are:

Small elliptical path

When the tangential velocity of an object is less than required for a circular orbit, the object will follow a small elliptical path.

0 < vT < √[G(M + m)/R]

If vT = 0, only the gravitational force along the radial axis would affect the falling object, such that the objects would collide. If vT was slightly greater than zero and both objects were point-masses, the falling object would travel in an elliptical orbit, just missing the other object.

However, if the objects have substantial size, it is possible that the orbiting object will collide with the fixed object. An example of this can be seen with Newton's Cannon, where the cannonball is not propelled fast enough to go into orbit.

(See Gravity and Newton's Cannon for more information.)

Cannonball does not have enough velocity to go into orbit

Cannonball does not have enough velocity to go into orbit

Also, since the mass of the cannonball is so small compared to the mass of the Earth, the approximation can be made:

0 < vT < √(GM/R)

Circular orbit

At a specific tangential velocity for the given masses and separation, one object will go in a circular orbit with respect to the other object according to the equation:

vT = √[G(M + m)/R]

If M >> m:

vT = √(GM/R)

As noted before, we assume there is no initial radial velocity that would complicate things and affect the shape of the orbit.

Large elliptical orbit

If the velocity of the object is greater than required for a circular object but is less than that of a parabolic path, the object will travel in a large elliptical orbit around the other object:

√[G(M + m)/R] < vT < √[2G(M + m)/R]

If M >> m:

√(GM/R) < vT < √(2GM/R)

Go out into space

If the velocity is great enough, the object will escape the gravitational force and fly out into space. It will travel in a parabolic or hyperbolic path, depending on the initial tangential velocity.

Parabolic path

If the tangential velocity is such that:

vT = √[2G(M + m)/R]

the object will follow a parabolic path and go off into space. This velocity is called the gravitational escape velocity.

(See Overview of Gravitational Escape Velocity for more information.)

Likewise, if M >> m:

vT = √(2GM/R)

Rocket follows parabolic path

Rocket follows parabolic path

Hyperbolic path

If the tangential velocity is is greater than required for a parabolic path, the object will follow a hyperbolic path and go off into space:

vT > √[2G(M + m)/R]

If M >> m:

vT > √(2GM/R)

In this situation, the velocity exceeds the escape velocity.

Summary

The tangential velocity of one object with respect to another determines whether the object collides with the other object, goes into orbit or flies off into space. The velocity equations for circular orbits and parabolic paths are the starting points for establishing the effect of tangential velocity on orbital motion.

The velocity equations are:

Collide or small elliptical orbit: 0 < vT < √[G(M + m)/R]

Circular orbit: vT = √[G(M + m)/R]

Large elliptical orbit: √[G(M + m)/R] < vT < √[2G(M + m)/R]

Parabolic path: vT = √[2G(M + m)/R]

Hyperbolic path: vT > √[2G(M + m)/R]

If the mass of one object is much greater than the other, the equations are simplified.


Be observant and curious about things


Resources and references

Ron Kurtus' Credentials

Websites

Circular Orbits - Wikipedia

Center of Mass Calculator - Univ. of Tennessee - Knoxville (Java applet)

Center of Mass - Wikipedia

Gravitation Resources

Books

Top-rated books on Gravity

Top-rated books on Gravitation


Questions and comments

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.


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