Motion — Question 9

Question 9

Testing the direction a one-dimensional model omits

A cart is modeled as moving on a prescribed straight laboratory axis. Its tracked marker may in reality wander sideways. Let $y$ be signed perpendicular distance from that fixed axis, in millimetres. The axis is fixed before the experiment; it is not fitted afterward to the data.

For one task, neglecting sideways motion is acceptable only if $|y|\leq0.50\ \mathrm{mm}$ throughout the run. At three instants an independent side-view measurement gives: The three successive transverse readings are $0.10,0.40,0.80\ \mathrm{mm}$; each has a hard error bound of $\pm0.20\ \mathrm{mm}$. Each true value can be anywhere within its stated interval. No probability distribution or bound on unobserved excursions is supplied.

1. For each sample, find the allowed interval for $y$. Classify it as fully within the task’s allowed band, overlapping its boundary, or wholly outside it.

2. Can this run satisfy the stated $0.50\ \mathrm{mm}$ criterion? Explain why one observation can reject a whole-run claim even though three observations cannot normally establish it.

3. A second task permits $|y|\leq1.00\ \mathrm{mm}$. What do the same samples establish for that task, and what remains unverified?

4. Does failure of the first task’s criterion prove the cart has no useful one-dimensional description for any purpose? State a sensible model or measurement change, without inventing a sideways-motion bound absent from the data.

Question 9 - Solution

Testing the direction a one-dimensional model omits

(a) Compare the entire measurement interval with the allowed band.

$$I_j=[y_j-0.20,y_j+0.20]:\quad I_1=[-0.10,0.30],\ I_2=[0.20,0.60],\ I_3=[0.60,1.00]\ \mathrm{mm}.$$

The original task allows $y\in[-0.50,0.50]\ \mathrm{mm}$. The measurement intervals are $$\boxed{[-0.10,0.30],\quad[0.20,0.60],\quad[0.60,1.00]\ \mathrm{mm}}.$$ The first lies fully inside the allowed band. The second overlaps its positive boundary: some allowed true values pass and others fail. The third is wholly outside, because even its smallest value, 0.60 mm, exceeds 0.50 mm.

(b) A single established violation rejects an all-times claim.

$$B=[-0.50,0.50]\ \mathrm{mm};\qquad I_1\subset B,\ I_2\cap B\ne\varnothing,\ I_3\cap B=\varnothing.$$

The third sample guarantees failure at that instant, so the run cannot satisfy the first task’s criterion throughout. This conclusion holds for every value allowed by that error bound.

By contrast, success at several samples does not exclude an unobserved excursion between them. Rejecting a claim that must hold everywhere requires one counterexample; establishing it requires coverage or a model controlling the unsampled intervals.

(c) A different tolerance leads to a different assessment.

$$I_1\cup I_2\cup I_3\subset[-1,1]\ \mathrm{mm}\quad\not\Rightarrow\quad |y(t)|\leq1\ \mathrm{mm}\ \forall t.$$

All three intervals lie inside $[-1.00,1.00]\ \mathrm{mm}$, including the permitted boundary value 1.00 mm. The second task’s criterion is therefore satisfied at all three sampled instants. It remains unverified between them. No maximum excursion or rate of sideways change can be deduced from the supplied data alone.

(d) Approximation quality depends on the question.

Failure at a 0.50 mm tolerance does not forbid a useful one-dimensional description at a looser tolerance or for another observable. For the original task, one could retain a transverse coordinate and measure it, improve the physical guide, or monitor the omitted motion with coverage and bounds appropriate to the full run. Merely labeling the track “straight” does not demonstrate the neglected error is small enough.