Motion — Question 6

Question 6

Continuity forces some events even when the path is unobserved

A point-like marker remains on a fixed straight rail. Three stationary gates are ordered A, B, C from left to right. The marker’s location varies continuously: it cannot jump over intervening places or leave the rail.

At four successive instants it is:

1. left of A;

2. right of C;

3. between A and B;

4. right of C again.

A crossing means passage from one side of a gate to the other, not merely touching the gate and returning to the same side. No marker lies exactly at a gate at a sampled instant. Arbitrary extra motion between samples is allowed.

1. List the gate crossings that are unavoidable in each of the three inter-sample intervals, in their required travel order. Give the minimum total count at each gate and over all gates.

2. What minimum number of changes between leftward and rightward travel must have occurred during the full record? Do pauses remove that requirement?

3. Are the minimum counts necessarily the actual counts? Describe how an unobserved excursion could increase them without changing any sample.

4. A supposedly complete detector log reports only two crossings of B during the full record. Can that log and the stated observations all be correct under the model? Identify assumptions that should be checked if such a discrepancy occurs experimentally.

Question 6 - Solution

Continuity forces some events even when the path is unobserved

(a) A continuous path must pass the intervening gates.

$$(ABC)\,|\,(CB)\,|\,(BC)\quad\Longrightarrow\quad(N_A,N_B,N_C)_{\min}=(1,3,3).$$

From the first to the second sample, the marker must cross $\boxed{A,\ B,\ C}$ in that order on a left-to-right traversal. From the second to the third, it must cross $\boxed{C,\ B}$ on a right-to-left traversal. From the third to the fourth, it must cross $\boxed{B,\ C}$ left to right.

These are unavoidable ordered crossings within each interval; extra crossings may occur around them. The minimum counts are $$\boxed{A:1,\qquad B:3,\qquad C:3,\qquad\text{total}:7}.$$ A path that travels monotonically between each pair of samples attains these minima.

(b) The required travel directions alternate.

$$N_{\mathrm{crossings}}\geq1+3+3=7,\qquad N_{\mathrm{reversals}}\geq2.$$

The three transitions require rightward, then leftward, then rightward travel. At least $\boxed{2}$ changes of travel direction are therefore needed. Waiting can separate these motions, but cannot turn a required leftward traversal into a rightward one.

(c) Samples impose lower bounds, not a complete event list.

For a finite crossing log, every added outward-and-return excursion contributes an even number of crossings: $$N_B=3+2k\quad(k=0,1,2,\ldots),\qquad N_B=2\text{ is impossible}.$$

For example, while ending the second interval between A and B, the marker could make an extra excursion across A and return across it. This adds two crossings of A while preserving all four sampled locations. Similar back-and-forth excursions can add crossings elsewhere. The data do not supply a finite maximum count.

(d) Check consistency before inventing a detailed path.

Two total crossings of B are fewer than the required three, so the detector log contradicts the observations and model. Possible failures include missed detector events, incorrect sample order or object identity, incorrect gate labels, or a path that leaves the assumed rail. No continuous history on that rail can repair the contradiction while keeping every stated datum valid.