Position and Displacement — Question 10

Question 10

Extrema impose a stronger distance bound than endpoints alone

A continuous path on a fixed east-positive line starts at $x_i=2\ \mathrm{m}$ and ends at $x_f=8\ \mathrm{m}$. During the run its smallest position is exactly $-1\ \mathrm{m}$ and its largest is exactly $11\ \mathrm{m}$. Both extrema are reached; the path never leaves that range. Their first-visit order is initially unknown.

All measurements are exact in this model. Distance means total length actually traveled, counting both directions. The timing of the motion is irrelevant to the questions below.

1. Find the net displacement and the lower bound on distance that would follow from the endpoints alone.

2. Find the minimum possible distance if the minimum position is reached for the first time before the maximum position. Then find the minimum if the maximum is first reached before the minimum. Exhibit a path attaining each bound.

3. An odometer reports a distance of exactly 22 m. Which first-visit order is possible? Construct a path with that distance, correct endpoints, and correct extrema.

4. Does the inferred first-visit order establish a unique path, or even establish that the initial travel was westward? Use a suitable example to explain.

5. Would a distance report of 16 m be compatible with all the endpoint and extremum data? Explain why merely checking $D\geq|\Delta x|$ is not sufficient here.

Question 10 - Solution

Extrema impose a stronger distance bound than endpoints alone

(a) Endpoints determine displacement and a basic distance bound.

$$\Delta x=8-2=\boxed{+6\ \mathrm{m}},\qquad D\geq|\Delta x|=6\ \mathrm{m}.$$ This basic bound does not yet use the required visits to the extrema.

(b) Add the shortest necessary legs in each first-visit order.

$$D_{\min\,\mathrm{first}}\geq|{-1}-2|+|11-(-1)|+|8-11|=18\ \mathrm{m}.$$

If the minimum is first, any path must include the ordered visits $2\to-1\to11\to8$. Their separations give $$D\geq3+12+3=\boxed{18\ \mathrm{m}}.$$ Monotone travel on those three legs attains the bound and stays within the range.

If the maximum is first, the necessary order is $2\to11\to-1\to8$, giving $$D\geq9+12+9=\boxed{30\ \mathrm{m}}.$$ Again, three monotone legs attain the bound. Extra excursions cannot reduce either required length.

(c) Use the measured distance to eliminate an order.

$$D_{\max\,\mathrm{first}}\geq|11-2|+|{-1}-11|+|8-(-1)|=30\ \mathrm{m}.$$

Since $18\leq22<30$, only first reaching the minimum before the maximum is possible. One compatible route is $$\boxed{2\to4\to-1\to11\to8\quad(\mathrm{m})},$$ whose length is $2+5+12+3=22\ \mathrm{m}$. Its extrema are exactly $-1$ and 11 m.

(d) An extremum order is not a complete route history.

The example above initially moves east to 4 m, even though it reaches the minimum before the maximum. Another route is $2\to-1\to1\to-1\to11\to8$, with length $3+2+2+12+3=22\ \mathrm{m}$. It initially moves west. Both satisfy every measurement, so neither the full path nor its initial direction is uniquely determined.

(e) Enforce all constraints, not just endpoint separation.

A 16 m report satisfies $16\geq6$ but is below the stronger 18 m minimum required by visiting both extrema. It is therefore incompatible with the complete data. Endpoint displacement is only one constraint on traveled distance.