Center of Mass Motion Components
by Ron Kurtus
The motion vectors of two objects in space with respect to the center of mass (CM) between them can be broken into their radial and tangential components. One reason to break the motion into its components is to facilitate calculations, especially since radial motion is affected by the gravitational attraction between the objects, while tangential speed in unaffected by gravitation.
When viewed from the CM, the objects move in the radial direction either both toward the CM or both away from the CM. The objects move in opposite directions tangential to the line through the CM.
The combination of the radial and tangential motion components determine whether the objects will collide, go into orbit or fly off into space.
Questions you may have include:
- How are the motion vectors broken into components?
- What are the radial components of motion?
- What are the tangential components of motion?
This lesson will answer those questions. Useful tool: Units Conversion
Motion broken into components
When two objects in space are viewed as moving with respect to the CM between them, their motion can be broken into radial and tangential vectors in the coordinate system, with the CM as the fixed-axis point.
Motion of objects can be broken into components
The radial vector components are on the axis through the CM. The tangential components are perpendicular to that axis.
When viewed with respect to the CM, the velocity vectors follow the ratio:
mvm = −MvM
This expression simply means that the velocity vectors are in opposite directions, when viewed from the CM. It is not to be confused with an overall direction convention.
The radial velocity vector is:
vR = v*cos(θ)
- vR is the radial velocity vector
- v is the velocity of the object
- θ (Greek letter theta) is the angle
- cos(θ) is the cosine of angle theta
Note: The angle is measured from the radial axis to the velocity vector, in a counterclockwise direction.
When θ > 90°, cos(θ) is negative. Thus vr is pointing away from the CM and is also negative.
Note: This follows that mvm = −MvM means the vectors are in opposite directions.
The tangential velocity vector is:
vT = v*sin(θ)
- vT is the tangential velocity vector
- sin(θ) is the sine of angle theta
When θ > 180°, sin(θ) is negative. Thus vT is pointing in the opposite direction and is also negative.
The radial vectors are along the axis between the objects. When viewed with respect to the CM, their motion is either toward the CM or away from the CM. The radial velocities between the objects are indicated as vRm and vRM.
Radial vector components of the two objects
Note: If the objects appear to be moving in the same direction, such that one object appears to be moving toward the CM, while the other is moving away from the CM, the perspective is not with respect to the CM. Changing the point of view, corrects the apparent motion.
Assuming the tangential velocity is zero, the objects will only move on the radial axis, according to the initial direction of their velocity and the effect of gravitation. The addition of tangential motion to the objects can cause then to go into orbit or fly off into space.
Motion away from CM
If the objects are moving away from the CM, they may reach a maximum displacement and then change directions and fall back toward each other and the CM. This is similar to the case of throwing a ball upward and having it return to Earth.
(See Overview of Gravity Equations for Objects Projected Upward for more information.)
If the initial radial velocity in a direction away from the CM is sufficient, the objects can escape from their gravitational attraction.
(See Center of Mass and Radial Gravitational Motion for more information.)
A tangential vector is perpendicular to the radial component and the axis between the objects. The tangential velocities of the objects are indicated as vTm and vTM. The tangential velocities are in opposite directions when viewed relative to the CM.
Tangential vector components of the two objects
Since there is always motion in the radial direction, the combination of the tangential and radial velocities will determine whether the objects will move past each other and go off into space, go into orbit around the CM or collide.
(See Center of Mass and Tangential Gravitational Motion for more information.)
Amount of tangential velocities
The combination of the tangential velocities and the gravitational attraction toward the CM, cause the objects to move in curved paths about the CM.
Depending on the tangential velocity, the objects will move in elliptical or circular orbits. If the velocity is great enough, the objects will go into parabola or hyperbolic paths and escape into space.
If there are no tangential velocities, the objects will move inward and collide at the CM.
The gravitational motion of two objects in space can be viewed with respect to the center of mass (CM) between them.
The motion vector of each object can be broken into its radial and tangential components, with respect to that CM. Since there is always a gravitational attraction between the objects, there is always a radial component to their motion. The combination of the radial and tangential motion components determine whether the objects will collide, go into orbit or fly off into space.
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Resources and references
Center of Mass Calculator - Univ. of Tennessee - Knoxville (Java applet)
Center of Mass - Wikipedia
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Center of Mass Motion Components