Motion — Question 7

Question 7

Following a dot requires knowing which object it is

A camera records two visually identical point markers on a fixed line. It measures location perfectly but cannot identify which marker produced a dot.

At $t=0$, call the left marker A and the right marker B. Both paths are continuous. Initially allow the ideal markers to pass through one another without interaction; they may occupy the same location between exposures. This is a tracking model, not a model of colliding solid bodies.

1. Construct one assignment of the dots in which each marker returns to its own starting location at $t=2\ \mathrm{s}$.

2. Construct a different continuous history in which the markers exchange starting locations by $t=2\ \mathrm{s}$. Show that all three unlabeled images are unchanged.

3. If the markers are instead impenetrable objects confined to a single narrow rail where passing is impossible, which assignment at the final instant remains possible? Does that imply the objects never moved?

4. Suggest one observational change that distinguishes the two ideal-marker histories at the final exposure. Would that change alone determine everything each marker did between exposures?

Question 7 - Solution

Following a dot requires knowing which object it is

(a) Assign identities consistently to a returning history.

$$\text{return: }A:0\to1\to0,\ B:4\to3\to4;\qquad\text{exchange: }A:0\to1\to4,\ B:4\to3\to0.$$

One possible record of locations, in metres, is $$A:0\ \longrightarrow\ 1\ \longrightarrow\ 0,\qquad B:4\ \longrightarrow\ 3\ \longrightarrow\ 4.$$ Continuous travel between those locations makes each marker return to its own start.

(b) Another identity assignment gives an exchange.

$$x_A=1+3(t-1),\quad x_B=3-3(t-1)\quad\Rightarrow\quad t=\frac43\ \mathrm{s},\ x=2\ \mathrm{m}.$$

A different record is $$A:0\ \longrightarrow\ 1\ \longrightarrow\ 4,\qquad B:4\ \longrightarrow\ 3\ \longrightarrow\ 0.$$ Continuous straight segments between successive timed locations provide one explicit history. In the second interval the markers meet and pass; this is allowed by the ideal model. Each image still contains exactly the same unordered pair of locations: $\{0,4\}$, then $\{1,3\}$, then $\{0,4\}$.

(c) A no-passing constraint preserves left–right order.

$$x_A(t)<x_B(t)\text{ throughout}\quad\Rightarrow\quad x_A(2)=0,\ x_B(2)=4\ \mathrm{m}.$$

If passing is physically impossible on the single rail, A must remain the left object and B the right object. Therefore the final assignment is A at 0 m and B at 4 m. The exchanged assignment is excluded because it reverses their order.

The middle image nevertheless establishes motion: the objects were at 1 and 3 m then. Return to a starting arrangement is not rest throughout the interval.

(d) Measure identity as well as location.

Give one marker a distinct visible label, color, or shape that remains recognizable. The final image would then reveal whether A is at 0 or 4 m, distinguishing the two proposed histories. A tag does not reveal unrecorded excursions between exposures. Identity and temporal coverage are separate requirements for reconstructing motion.