Find what a straight sampled track does not reveal

Begin with a measurement question

A tracking table provides positions at particular times. The difference between neighboring positions divided by elapsed time is an interval-average velocity. A straight line through the recorded points is consistent with constant velocity, but it does not prove that every unrecorded instant followed that line. The information between samples is an inference, not another measurement.

Build the algebraic model

The workbench deliberately supplies two different tracks with the same recorded frames. One is x=2t. The other is x=2t+A sin(2πt/Δt), where Δt is the recording interval and A is a hidden oscillation amplitude. At integer multiples of Δt, the sine term is zero, so both tracks give the same five recorded positions. This is an invented ambiguity demonstration, not a claim that a real object necessarily oscillated.

Check a concrete implication

For sufficiently large amplitude, the revealed track reverses between frames. Its displacement still equals final minus initial position, but its path distance must add every forward and backward segment. The software locates the known generator’s turning points to compute that distance. Those turning points are visible because the generator is disclosed; they are not uniquely recovered from the five samples.

Challenge the evidence boundary

At Δt=0.5 s, all displayed interval velocities are 2 m/s. With A=0 the path distance over four intervals is 4 m. With A=0.5 m there are eight between-frame reversals and a longer path, even though recorded endpoints remain unchanged. A new sampling schedule not synchronized to the hidden period can distinguish these two hypotheses. Faster sampling alone is not magic: its timing and resolution must actually resolve the proposed alternative.

Worked original case

At Δt=0.5 s, the five records are (0,0),(0.5,1),(1,2),(1.5,3),(2,4) in seconds and metres for every selected amplitude. All secants are 2 m/s, while the disclosed oscillatory track can have greater average speed.

Predict and investigate

Do equal neighboring secant velocities prove constant instantaneous speed? Compare amplitudes 0,0.25,0.5 m at one sampling interval. Retain the identical recorded points and different generator distances. State which extra observation time would distinguish constant motion from the selected hidden oscillation without treating the revealed curve as measured data.

Changed numerical practice

Try the changed case before feedback

A tracker records x=1 m at t=0 and x=7 m at t=3 s. Find average velocity in m/s. Can these two readings establish average speed?

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Reflect and source-check

Why is a table interpolating a known generator not independent evidence for that generator?

Original explanations, synthetic records and models; checked 30 October 2026 against OpenStax College Physics 2e. The guided calculation is algebra-based; no calculus prerequisite is assumed. All activity is virtual. No physical launch, drop, water crossing, machinery operation or medical decision is requested. A public answer or saved capture does not verify independent mastery or actual measurement technique.