Motion — Question 4

Question 4

Equal dot spacing needs a clock before it describes pace

A marker is photographed at four locations on a fixed straight rail:

Two candidate timestamp records are proposed for the same four pictures: For exposures A, B, C and D respectively, record I gives times $0,1,2,3\ \mathrm{s}$, whereas record II gives $0,1,3,6\ \mathrm{s}$. Treat locations and timestamps as exact. For this question, “steady forward pace” means continuous forward travel with equal lengths covered in equal durations throughout the observed run. No speed formula is needed.

1. What spatial fact is shared by both records? Find the three elapsed times between successive exposures in each record.

2. Which timestamp record is compatible with a steady forward pace throughout? Which rules it out? Explain by comparing the equal spatial gaps with their elapsed times.

3. Does the compatible record prove steady travel between pictures? Describe a continuous alternative that pauses during every inter-picture interval but gives the same four samples.

4. If all timestamps are removed but the exposure order is kept, which conclusion about changing pace can still be made from the equal spacing alone? Identify the missing information.

Question 4 - Solution

Equal dot spacing needs a clock before it describes pace

(a) Equal spatial gaps can have unequal durations.

$$(\Delta t_{AB},\Delta t_{BC},\Delta t_{CD})_I=(1,1,1)\ \mathrm{s},\quad(\cdots)_{II}=(1,2,3)\ \mathrm{s}.$$

Every consecutive sampled location is 2 m farther along the rail. Record I gives elapsed times $\boxed{1,1,1\ \mathrm{s}}$. Record II gives $\boxed{1,2,3\ \mathrm{s}}$.

(b) Compare like lengths with the durations they took.

$$\Delta x_{AB}=\Delta x_{BC}=\Delta x_{CD}=2\ \mathrm{m};\quad\text{equal gaps alone do not fix equal times.}$$

Record I is compatible with continuous steady forward travel: each equal length takes an equal duration. Record II rules out that description over the whole run, because its equal 2 m gaps take successively longer durations. This conclusion requires no calculation of a numerical speed.

(c) Compatibility is not proof of the intervening history.

Using numerical seconds and metres, a continuous stop–move counterexample is $$x(t)=2k+\begin{cases}0,&0\leq t-k\leq\tfrac12,\\4(t-k-\tfrac12),&\tfrac12<t-k\leq1,\end{cases}\quad k=0,1,2.$$

For Record I, a marker could remain at each sampled location for the first half of the next one-second interval, then move continuously forward by 2 m during the second half. It would reach the next sampled mark on time and repeat. This history has pauses and is not steady throughout, yet matches all four images.

The idealized changes between waiting and travel need not be treated as a dynamics model. Smooth departures and arrivals can also be arranged inside each interval without changing the sampled positions. The pictures alone do not select a unique history.

(d) A spacing pattern without timing cannot establish changing pace.

Without timestamps or a verified equal-exposure-interval setting, equal dot spacing alone neither proves steady pace nor proves a change of pace. Both timestamp records fit the same spatial pattern. Ordering tells which recorded location came next; calibrated timing tells how long the transitions took.