Position and Displacement — Question 2

Question 2

Changing the origin and reversing the axis are different operations

A marker has initial and final coordinates $x_i=-4\ \mathrm{m}$ and $x_f=9\ \mathrm{m}$ on a fixed east-positive axis.

A surveyor introduces another fixed origin at $x=12\ \mathrm{m}$. First she defines an east-positive coordinate $u$ from that origin. She then defines a west-positive coordinate $y$ from the same new origin. All coordinates use metres.

1. Derive the coordinate relations $u(x)$ and $y(x)$ by checking the new origin and the direction in which each coordinate increases.

2. Find both endpoint coordinates and the displacement in each of the $x$, $u$, and $y$ conventions.

3. Explain why moving a fixed origin leaves the displacement unchanged when axis orientation is retained, while reversing the axis changes its sign. What physical description remains the same?

4. A report states only “displacement $=-13\ \mathrm{m}$.” Can a reader conclude the marker moved net westward without knowing the coordinate convention? What information should the report include?

Question 2 - Solution

Changing the origin and reversing the axis are different operations

(a) Enforce the new zero and the declared orientation.

The east-positive coordinate must vanish at $x=12\ \mathrm{m}$ and increase when $x$ increases: $$\boxed{u=x-12\ \mathrm{m}}.$$ The west-positive coordinate also vanishes there but decreases when $x$ increases: $$\boxed{y=12\ \mathrm{m}-x=-u}.$$

(b) Transform the endpoints before taking differences.

$$\Delta u=(x_f-12)-(x_i-12)=\Delta x,\qquad\Delta y=(12-x_f)-(12-x_i)=-\Delta x.$$

$$\begin{aligned}(x_i,x_f)&=(-4,9)\ \mathrm{m},&\Delta x&=+13\ \mathrm{m},\\(u_i,u_f)&=(-16,-3)\ \mathrm{m},&\Delta u&=+13\ \mathrm{m},\\(y_i,y_f)&=(16,3)\ \mathrm{m},&\Delta y&=-13\ \mathrm{m}.\end{aligned}$$

(c) A fixed offset cancels; an orientation reversal does not.

For an arbitrary fixed shift $a$, $(x_f-a)-(x_i-a)=x_f-x_i$. In contrast, subtracting the reversed endpoint coordinates gives $\Delta y=-(x_f-x_i)=-\Delta x$.

All three descriptions represent the same physical endpoint change: $\boxed{13\ \mathrm{m}\text{ east}}$. The magnitude is unchanged. A negative displacement in the west-positive convention denotes a net change toward the east, not a different motion.

(d) State the coordinate convention with a signed answer.

$$\Delta\mathbf r=(13\ \mathrm{m})\hat{\mathbf e}_{E}=(-13\ \mathrm{m})\hat{\mathbf e}_{W}.$$

The bare value $-13\ \mathrm{m}$ does not identify east or west until the positive direction is specified. A useful report identifies the reference, positive direction, and selected initial/final events, or states the physical direction directly. Signed coordinates are a description of the geometry; changing their convention does not move the object.