Relative Position, Velocity, and Acceleration in One Dimension


Quick
The relative velocity is the velocity seen by a particular observer, or the velocity relative to that observer.
Each observer, equipped in principle with a measuring stick and a stopwatch, forms a frame of reference.


Equations
(Eq1)    
xB = xA + xB/A
relative position
(Eq2)    
vB = vA + vB/A
relative velocity
(Eq3)    
aB = aA + aB/A
relative acceleration


Nomenclature
xB/Arelative position of B with respect to A
vB/Arelative velocity of B with respect to A
aB/Arelative acceleration of B with respect to A


Details

In general, when two observers measure the velocity of a moving body, they get different results if one observer is moving relative to the other. The velocity seen by a particular observer is called the velocity relative to the other. The velocity seen by a particular observer is called the velocity relative to that observer, or simply relative velocity. For straight-line motion the term velocity is used to mean the component of the velocity vector along the line of motion; this can be positive, negative, or zero.

Consider two particles A and B moving along the same straight line as shown:



If the position coordinates xA and xB are measured from the same origin, the difference xBxA defines the relative position coordinate of B with respect to A and is denoted by xB/A. Therefore:

xB/A = xBxA

or:

(Eq1)    
xB = xA + xB/A

Regardless of the positions of A and B with respect to the origin, a positive sign for xB/A means that B is to the right of A, and a negative sign means that B is to the left of A.

The rate of change of xB/A is known as the relative velocity of B with respect to A and is denoted by vB/A. Differentiating Eq1 to obtain velocity:

vB/A = vBvA

or:

(Eq2)    
vB = vA + vB/A

A positive sign for vB/A means that B is observed from A to move in the positive direction; a negative sign means that it is observed to move in the negative direction.

The rate of change of vB/A is known as the relative acceleration of B with respect to A and is denoted by aB/A. Differentiating Eq2 to obtain acceleration:

aB/A = aBaA

or:

(Eq3)    
aB = aA + aB/A

Note that the product of the subscripts A and B/A used in the right-hand member of Eq1, Eq2, and Eq3 is equal to the subscript B used in their left-hand member.