Motion as a Change in Position

Comparing Positions at Different Times

An object is moving relative to a frame of reference when its position changes as time passes. One observation is therefore not enough. Compare its positions at several times to determine whether it moved. The initial position and final position summarize its net change over one interval.

If the initial position is $x_{\text{initial}}$ and the final position is $x_{\text{final}}$ , the change in position is:

\Delta x=x_{\text{final}}-x_{\text{initial}}

The symbol $\Delta x$ is read as "delta x" and means displacement on this line. Its value can be positive, negative, or zero.

Motion is identified from position changing with time. Displacement compares only the two chosen endpoints.

Position Difference on a Straight Track

On a straight track with a fixed zero point, position change depends only on final position minus initial position. For example, if an object starts at $20 \text{ m}$ and ends at $50 \text{ m}$ , the change in position is $+30 \text{ m}$ .

If the initial and final positions are the same, the displacement is zero. Those two endpoints alone do not tell us whether the object stayed still or moved away and returned. Intermediate positions or the traveled path are needed to distinguish those cases.

Keep the Reference Consistent

When calculating a change in position, the initial and final positions must be read from the same reference point. Do not take the initial position from the store and the final position from another house, because that no longer describes the same motion in one reference frame.

Reference	Initial position	Final position	Change in position
Store at $0 \text{ m}$	$+20 \text{ m}$	$+50 \text{ m}$	$+30 \text{ m}$
Rani's house at $20 \text{ m}$	$0 \text{ m}$	$+30 \text{ m}$	$+30 \text{ m}$
Naya's house at $50 \text{ m}$	$-30 \text{ m}$	$0 \text{ m}$	$+30 \text{ m}$

The three rows have different initial and final position numbers, but the same change in position. The reference point shifts the coordinate numbers. The final-minus-initial difference still describes the motion from start to finish.

Reading the Sign of Position Change

On a straight line, we may choose one direction as positive. In this example, east is positive.

$\Delta x>0$ means the final position is east of the initial position.
$\Delta x<0$ means the final position is west of the initial position.
$\Delta x=0$ means the initial and final positions are the same.

A negative sign does not mean the motion is smaller or wrong. It only tells the direction of the position change.

Subtract the Final and Initial Positions

A student starts $10 \text{ m}$ east of a gate. A few seconds later, the student is $40 \text{ m}$ east of the gate. The change in position is:

\begin{aligned} \Delta x &=x_{\text{final}}-x_{\text{initial}} \\ &=40 \text{ m}-10 \text{ m} \\ &=+30 \text{ m} \end{aligned}

So, the student moves $30 \text{ m}$ to the east. We know motion happened because the final position is different from the initial position.

Notice the order of subtraction. We always read final position minus initial position, then use the sign of the result to decide the direction of the position change.