# Relative Motion and Center of Mass

by Ron Kurtus (14 May 2011)

The motion of two objects in space is relative to some defined point of reference. Typically, that point of reference is the center of mass (CM) between the objects.

However, you may also observe the motion with respect to your own viewpoint or some fixed reference. In such a case, you often cannot tell where the center of mass (CM) is actually located.

Another point of reference is that the motion can be with respect to one of the two objects, such as viewing the motion of the Moon with respect to the Earth.

Questions you may have include:

• What is the motion relative to an outside observer?
• What is the motion relative to the center of mass?
• What is the motion relative to one of the objects?

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

## Motion relative to outside observer

When you observe two moving objects in space, your point of view concerning that motion usually is that you are stationary and the objects are moving with respect to you. This is especially evident when both objects are moving in the same direction.

Your viewpoint of the motion of the center of mass (CM) is a function of the velocities of the objects relative to you or to some other stationary object and their masses. The general velocity relationship is:

vCM= (mvm + MvM)/(m + M)

where

• m and M are the masses of the objects
• vCM is the velocity of the CM
• vm and vM are the velocities of the objects

The system of two objects appears to be moving, and it is difficult to tell where their CM is located.

## Motion relative to CM

Most often, the motion of two objects in space is illustrated as with respect to a stationary CM. This is the ideal case. For example, when you observe two objects on orbit around their CM—such as with twin stars—you can usually see them orbit the fixed CM between them. Your view is relative to that CM, such that vCM = 0.

The general velocity equation becomes:

0 = (mvm + MvM)/(m + M)

mvm = −MvM

This means that the velocity vectors are in opposite directions, when viewed from the CM. In this ideal case, it can be said that the motions of the objects mirror each other, according to their respective masses. This "mirroring" can be seen when both objects move apart, move toward each other, go into orbit and fly off into space.

## Motion relative to one object

Instead of the two objects moving with respect to you or relative to the CM, the motion of one object is sometimes viewed with respect to the other object, as if it were stationary. An example of this is when you view the Moon's orbit from the Earth.

If the object of mass m is seems to be moving with respect to object M, its velocity is the difference in their velocity vectors with respect to the CM:

vmM = vm − vM

where

• vmM is the velocity vector of object m with respect to object M
• vm and vM are the velocity vectors of the two objects with respect to the CM

Since the velocity vectors are in opposite directions or mvm = −MvM, the magnitude of vmM is the sum of the magnitudes of the two velocities.

## Summary

The motion of two objects in space is relative to some fixed reference point. In many cases, you may see the motion with respect to you viewpoint. Ideally, the viewpoint is relative to a stationary CM between the two objects. It is also possible to view the motion with respect to one of the two objects.

You can move mountains with your thoughts

## Resources and references

Ron Kurtus' Credentials

### Websites

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

Center of Mass - Wikipedia

Gravitation Resources

### Books

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www.school-for-champions.com/science/
gravitation_center_of_mass_relative_motion.htm

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