Position and Displacement — Question 4

Question 4

Can total distance reveal where a traveler turned?

A traveler starts at $x_i=-3\ \mathrm{m}$ and ends at $x_f=7\ \mathrm{m}$ on a straight east-positive line. A reliable path-length recorder reports a total traveled distance of $34\ \mathrm{m}$. All data are exact in this model.

Let $R\geq0$ and $L\geq0$ be the total eastward and westward traveled lengths. For any such path, $D=R+L$ and $\Delta x=R-L$.

1. Find the displacement and solve for $R$ and $L$. Does a net eastward displacement mean that all travel was eastward?

2. Suppose there is exactly one change of direction. Find the turning-point coordinate for an east-first trip and for a west-first trip. Check the distance of each proposed route.

3. Do the measured endpoints and distance alone determine which turning point was visited? How does allowing more than one reversal affect uniqueness?

4. Add the independent condition that the traveler always remains within the corridor $-10\ \mathrm{m}\leq x\leq20\ \mathrm{m}$. Under the exactly-one-reversal assumption, which candidate survives? Does the corridor condition alone establish exactly one reversal?

Question 4 - Solution

Can total distance reveal where a traveler turned?

(a) Add and subtract the directional-length equations.

$$R=\frac{D+\Delta x}{2}=22\ \mathrm{m},\qquad L=\frac{D-\Delta x}{2}=12\ \mathrm{m}.$$

The displacement is $7-(-3)=\boxed{+10\ \mathrm{m}}$. Solving $R+L=34\ \mathrm{m}$ and $R-L=10\ \mathrm{m}$ gives $$\boxed{R=22\ \mathrm{m}},\qquad \boxed{L=12\ \mathrm{m}}.$$ Net eastward change coexists with 12 m of westward travel.

(b) Exactly one reversal permits two orderings.

$$2x_{\mathrm{turn}}-(x_i+x_f)=34\Rightarrow x_{\mathrm{turn}}=19\ \mathrm{m};\quad x_i+x_f-2x_{\mathrm{turn}}=34\Rightarrow x_{\mathrm{turn}}=-15\ \mathrm{m}.$$

For an east-first trip, the turn occurs at $-3+22=\boxed{19\ \mathrm{m}}$, followed by 12 m west to 7 m. Its length is $22+12=34\ \mathrm{m}$.

For a west-first trip, the turn occurs at $-3-12=\boxed{-15\ \mathrm{m}}$, followed by 22 m east to 7 m. Its length is $12+22=34\ \mathrm{m}$.

(c) Directional totals do not specify travel order.

Both candidates reproduce the measured data, so those data do not select one turning point. With multiple reversals, the same total eastward and westward lengths can be distributed among more legs, creating additional compatible paths. The recorder supplies length, not a complete route history.

(d) Apply the additional spatial constraint separately.

The 19 m turn lies inside the corridor, and its entire two-leg route stays within the limits. The $-15\ \mathrm{m}$ turn lies outside and is excluded. Thus the east-first candidate survives if exactly one reversal is known.

The corridor itself does not prove one reversal. For example, $$-3\ \longrightarrow\ 9\ \longrightarrow\ 7\ \longrightarrow\ 17\ \longrightarrow\ 7 \quad(\mathrm{m})$$ is inside the corridor and has length $12+2+10+10=34\ \mathrm{m}$, with the same endpoints but three reversals. Each assumption removes a different part of the ambiguity.