Reference Frame and Position

Position Requires a Reference Point

If you say, "I am $30 \text{ m}$ from the store," your friend can still ask, "which direction?" If you say, "I am east," your friend can still ask, "how far?"

In physics, position is an object's location in a chosen frame of reference. The frame provides a coordinate system and a positive direction. A store, house, flagpole, or start line can serve as the reference point, or origin, inside that frame.

A complete position statement names the reference point, value, unit, and direction.

If motion happens along one straight line, we can choose one direction as positive. In the street example, east is positive and west is negative.

x'=x_{\text{object}}-x_{\text{origin}}

This equation keeps the object in place and shifts the coordinate origin while the axis and positive direction stay fixed.

Position from Different References

Take one object at coordinate $30 \text{ m}$ when the store is used as zero. If the origin is shifted to Rani's house at coordinate $20 \text{ m}$ , the object's new coordinate becomes $+10 \text{ m}$ .

The object does not move when the origin changes. Only its coordinate changes. Shifting the origin to the right can turn a large positive coordinate into a smaller positive coordinate, zero, or a negative coordinate.

Reading Sign and Direction

The sign of a position tells us which side of the reference point the object is on. In the street example, east is chosen as positive. So a positive value means the object is east of the reference point, while a negative value means the object is west of it.

Origin in the same reference frame	Object coordinate on the street axis	Coordinate relative to the origin
Store at $0 \text{ m}$	Object at $30 \text{ m}$	$+30 \text{ m}$ east
Rani's house at $20 \text{ m}$	Object at $30 \text{ m}$	$+10 \text{ m}$ east
Naya's house at $50 \text{ m}$	Object at $30 \text{ m}$	$-20 \text{ m}$ or $20 \text{ m}$ west

The value $-20 \text{ m}$ does not mean the distance is negative. The negative sign only tells the direction relative to the reference point.

A Complete Position Statement

A position sentence needs four parts. If one part is missing, another person can read the object's location from the wrong reference point.

Building a Position Statement

A complete statement is: "The object is $10 \text{ m}$ east of Rani's house." It specifies the object, reference point, coordinate magnitude, unit, and direction.

The sentence "the object is near the house" omits the reference point, numerical distance, unit, and direction. Kinematics needs all four to assign a coordinate to the object.

A Position Sentence from the Store

A marker is at coordinate $40 \text{ m}$ when the store is the origin. Dimas's house is at coordinate $30 \text{ m}$ in the same frame. If the origin is shifted to Dimas's house, the marker's coordinate is:

\begin{aligned} x' &=x_{\text{marker}}-x_{\text{Dimas}} \\ &=40 \text{ m}-30 \text{ m} \\ &=+10 \text{ m} \end{aligned}

So, the marker is $10 \text{ m}$ east of Dimas's house. With the store as origin, its coordinate is $40 \text{ m}$ . With Dimas's house as origin, its coordinate is $+10 \text{ m}$ . Both statements are correct because they use different origins in a clearly defined reference frame.