SCIENCE · INVESTIGATE AND EXPLAIN

Distinguish an origin error from a failed projectile model

A model prediction and a sensor readout can disagree for different reasons. A constant horizontal origin error shifts every position by the same amount. A change in horizontal acceleration instead changes curvature over time. The supplied comparison is designed to distinguish those two explanations under explicitly exact timing and fixed spatial scale.

Learning outcome: Compare a=0 and1 at offset 0.1 m, then change offset to −0.2 m. Retain raw, corrected and residual coordinates. Predict the second difference before checking and propose a separate timing/perspective test that the packet cannot supply.

Starting recap

Use SI units, signed components, algebraic ratios and graph coordinates. Identify the measurement rule and distinguish a record from an interpolated model.

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Try

For a=1 and o=0.1 m, initial horizontal readout 0.1 m identifies the offset in this packet. The corrected second difference is −0.04 m over a 0.2 s step, giving acceleration −1 m/s². Origin correction leaves that curvature intact.

Open the experiment

Predict

Can subtracting the first position reading eliminate a true quadratic residual?

Your prediction and reason

Experiment

Compare a=0 and1 at offset 0.1 m, then change offset to −0.2 m. Retain raw, corrected and residual coordinates. Predict the second difference before checking and propose a separate timing/perspective test that the packet cannot supply.

Distinguish curvature from a position-origin error

Worked teaching model · synthetic values · Synthetic packet at0,.2,.4,.6,.8 s: x=4t−a t²/2+offset,y=2+2t−5t². Times and vertical scale exact. Constant horizontal deceleration is an assigned alternative, NOT a physical aerodynamic drag law; the displayed packet remains above ground.

Stipulated horizontal deceleration magnitude: 1 m/s² Horizontal readout offset: 0.1 m

Reset keeps your prediction, captures and practice history.

Raw and origin-corrected horizontal records

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: raw readings (open markers).2: origin-corrected readings (filled markers).3: no-horizontal-acceleration comparison (dotted). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Raw and origin-corrected horizontal records

raw readings

time (s)	horizontal coordinate (m)
0	0.1
0.2	0.88
0.4	1.62
0.6	2.32
0.8	2.98

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

origin-corrected readings

time (s)	horizontal coordinate (m)
0	0
0.2	0.78
0.4	1.52
0.6	2.22
0.8	2.88

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

no-horizontal-acceleration comparison

time (s)	horizontal coordinate (m)
0	0
0.2	0.8
0.4	1.6
0.6	2.4
0.8	3.2

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Residual after removing origin offset

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: observed minus constant-velocity comparison (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Residual after removing origin offset

observed minus constant-velocity comparison

time (s)	position residual (m)
0	0
0.2	-0.02
0.4	-0.08
0.6	-0.18
0.8	-0.32

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Indicated horizontal origin	0.1	m
Horizontal second difference	-0.04	m
Inferred signed horizontal acceleration	-1	m/s²
Final residual from constant4 m/s model	-0.32	m
Final height of sampled packet	0.4	m
Known generator horizontal deceleration	1	m/s²

A constant origin error shifts all positions, whereas the stipulated horizontal acceleration changes their curvature.

Second differences remove a constant offset and a linear velocity term. Under the exact timing/position assumptions, curvature rejects this no-horizontal-acceleration comparison, but does not identify physical drag. Perspective distortion, clock error, changing scale or other forces require independent checks. All five vertical positions remain above the ground; no post-impact data are fit.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

A constant origin error shifts all positions, whereas the stipulated horizontal acceleration changes their curvature.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Begin with a measurement question

Build the algebraic model

The synthetic packet uses x=4t−at²/2+o and y=2+2t−5t² at times 0,0.2,0.4,0.6,0.8 s. The parameter a is a stipulated constant horizontal deceleration magnitude, not an aerodynamic drag law. The origin offset o can be removed by subtracting the initial readout. That subtraction does not remove the quadratic term.

Check a concrete implication

After origin correction, compare the positions with the no-horizontal-acceleration prediction x=4t. Residual is−at²/2, which curves downward when a is positive. Equal-time second differences are−aΔt², so division by the squared time step gives signed horizontal acceleration. A residual pattern rejects the stated constant-horizontal-velocity comparison under the assumptions; it does not identify which physical force or imaging failure caused the discrepancy.

Challenge the evidence boundary

At a=1 m/s², the final residual at 0.8 s is −0.32 m regardless of offset. All five vertical points remain above ground, ending at 0.4 m, so the fit does not include post-contact positions. Perspective distortion, changing scale, clock error or unmodeled force could still mimic aspects of the pattern in a real record. Design an independent calibration or changed observation before naming a mechanism.

Worked original case

Predict and investigate

Can subtracting the first position reading eliminate a true quadratic residual? Compare a=0 and1 at offset 0.1 m, then change offset to −0.2 m. Retain raw, corrected and residual coordinates. Predict the second difference before checking and propose a separate timing/perspective test that the packet cannot supply.

Changed numerical practice

Try the changed case before feedback

At times 0,0.2,0.4 s, horizontal positions are0.1,0.88,1.62 m. Infer signed horizontal acceleration in m/s² assuming constant acceleration and exact times.

Numerical result in the requested units

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 does rejecting one ballistic assumption not by itself measure an air-drag coefficient?

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.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Why does rejecting one ballistic assumption not by itself measure an air-drag coefficient?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Find the landing event from the actual initial height Next: Challenge a force inventory with independent acceleration

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