Calibrate position before interpreting a speed estimate

Begin with a measurement question

A motion record begins with a measurement rule, not just a column of numbers. An instrument can have an origin offset, a scale error or both. In the supplied model, indicated position r=sx+o, where x is true position in metres, s is a dimensionless scale and o is a fixed offset in metres. Subtracting two indicated positions cancels o, but the difference remains s times the true displacement.

Build the algebraic model

Two known reference positions can determine this affine rule. With anchors at 0 and 2 m, indicated readings r0 and r2 give s=(r2−r0)/(2 m) and o=r0. A corrected coordinate is (r−o)/s. One anchor alone cannot determine both parameters. Converting centimetres into metres is a unit conversion; discovering that an instrument scale is wrong requires reference evidence, not merely relabeling its numbers.

Check a concrete implication

This target is stipulated to move monotonically in the positive direction from 0.4 to 2.8 m in 2 s. Its corrected average velocity and average speed therefore both equal 1.2 m/s; the two endpoints alone would not establish that equality if unseen reversals were allowed. A common origin shift changes both recorded positions equally and does not change the inferred raw displacement. A scale of 1.1 instead makes the raw speed 1.32 m/s. Repeated readings with little scatter can reproduce that same biased estimate: precision is not accuracy.

Challenge the evidence boundary

The separate raw-speed bounds allow ±0.01 m on each position reading and ±0.01 s on each time reading. Worst-case endpoint differences allow ±0.02 m and ±0.02 s, so divide the smallest allowed displacement by the largest allowed duration for the lower bound. These are deterministic bounds under explicit assumptions, not confidence intervals. The reference anchors are stipulated exact in the affine correction; a real calibration would also propagate their uncertainty.

Worked original case

At s=1.1 and o=0.1 m, anchor readings are 0.1 and 2.3 m. Target readings are 0.54 and 3.18 m, giving raw displacement 2.64 m and raw speed 1.32 m/s. Correcting scale recovers 2.4 m and 1.2 m/s.

Predict and investigate

Will subtracting the two position readings remove both offset and scale error? Compare scales 0.8,1.0,1.2 at offset 0.1 m, then change only offset. Retain anchor readings, raw and corrected speed and the stated raw interval. Propose an independent check for a nonlinear response that two anchors cannot exclude.

Changed numerical practice

Try the changed case before feedback

Reference positions 0 and 3 m give readings 0.2 and 3.8 m. A target changes indicated position by 2.4 m in 2 s. Find its corrected average velocity in m/s.

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

What assumption would fail if the indicated scale changed across the measurement range?

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.