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Relative Position, Velocity, and Acceleration in One Dimension

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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/A	relative position of B with respect to A
vB/A	relative velocity of B with respect to A
aB/A	relative 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 x_A and x_B are measured from the same origin, the difference x_B − x_A defines the relative position coordinate of B with respect to A and is denoted by x_B/A. Therefore:

x_B/A = x_B − x_A

or:

(Eq1)

xB = xA + xB/A

Regardless of the positions of A and B with respect to the origin, a positive sign for x_B/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 x_B/A is known as the relative velocity of B with respect to A and is denoted by v_B/A. Differentiating Eq1 to obtain velocity:

v_B/A = v_B − v_A

or:

(Eq2)

vB = vA + vB/A

A positive sign for v_B/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 v_B/A is known as the relative acceleration of B with respect to A and is denoted by a_B/A. Differentiating Eq2 to obtain acceleration:

a_B/A = a_B − a_A

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.