SCIENCE · INVESTIGATE AND EXPLAIN

Test a potential shape with evidence not used to fit it

Suppose the synthetic potential is U=x²+q x⁴+C. The constant C chooses an energy zero; it does not change force or energy differences. Fit a purely quadratic potential using only U(1)−U(0)=1+q J. This gives fitted stiffness2(1+q) N/m, but one energy difference cannot establish the whole spatial shape.

Learning outcome: Fit only the first difference, predict the withheld energy and independent force, then vary the common offset. Preserve which evidence was used and which was withheld.

Starting recap

Use vector dot products for work, gradients for conservative force, definite integrals for accumulation and explicit energy/frame boundaries. Check accessible states before assigning trajectories.

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Try

At q=.5, withheld energy residual is6 J and force residual−1 N. Changing C from−2 to+2 J changes neither residual nor force.

Open the experiment

Predict

A common additive potential offset changes:

Your prediction and reason

Experiment

Fit only the first difference, predict the withheld energy and independent force, then vary the common offset. Preserve which evidence was used and which was withheld.

Test a fitted potential with unused energy and force evidence

Worked teaching model · synthetic values · Synthetic U=x²+q x⁴+C J. Fit a purely quadratic potential using only the difference U(1)−U(0). Withhold U(2)−U(0) and separately supplied force at1 m. Common potential offset cannot affect force or energy differences; no measured material identification.

True quartic coefficient: 0.2 J/m⁴ Additive potential zero: 1 J

Reset keeps your prediction, captures and practice history.

Fitted potential shape and withheld evidence

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: supplied true shape (solid).2: quadratic fitted only at x1 (dashed). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Fitted potential shape and withheld evidence

supplied true shape

position (m)	potential energy relative to x0 (J)
0	0
0.025	0.0006250781
0.05	0.00250125
0.075	0.005631328
0.1	0.01002
0.125	0.01567383
0.15	0.02260125
0.175	0.03081258
0.2	0.04032
0.225	0.05113758
0.25	0.06328125
0.275	0.07676883
0.3	0.09162
0.325	0.1078563
0.35	0.1255012
0.375	0.1445801
0.4	0.16512
0.425	0.1871501
0.45	0.2107013
0.475	0.2358063
0.5	0.2625
0.525	0.2908188
0.55	0.3208013
0.575	0.3524876
0.6	0.38592
0.625	0.4211426
0.65	0.4582013
0.675	0.4971438
0.7	0.53802
0.725	0.5808813
0.75	0.6257812
0.775	0.6727751
0.8	0.72192
0.825	0.7732751
0.85	0.8269012
0.875	0.8828613
0.9	0.94122
0.925	1.002044
0.95	1.065401
0.975	1.131363
1	1.2
1.025	1.271388
1.05	1.345601
1.075	1.422719
1.1	1.50282
1.125	1.585986
1.15	1.672301
1.175	1.76185
1.2	1.85472
1.225	1.951
1.25	2.050781
1.275	2.154156
1.3	2.26122
1.325	2.372069
1.35	2.486801
1.375	2.605518
1.4	2.72832
1.425	2.855313
1.45	2.986601
1.475	3.122294
1.5	3.2625
1.525	3.407331
1.55	3.556901
1.575	3.711325
1.6	3.87072
1.625	4.035205
1.65	4.204901
1.675	4.379931
1.7	4.56042
1.725	4.746494
1.75	4.938281
1.775	5.135913
1.8	5.33952
1.825	5.549238
1.85	5.765201
1.875	5.987549
1.9	6.21642
1.925	6.451956
1.95	6.694301
1.975	6.9436
2	7.2

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.

quadratic fitted only at x1

position (m)	potential energy relative to x0 (J)
0	0
0.025	0.00075
0.05	0.003
0.075	0.00675
0.1	0.012
0.125	0.01875
0.15	0.027
0.175	0.03675
0.2	0.048
0.225	0.06075
0.25	0.075
0.275	0.09075
0.3	0.108
0.325	0.12675
0.35	0.147
0.375	0.16875
0.4	0.192
0.425	0.21675
0.45	0.243
0.475	0.27075
0.5	0.3
0.525	0.33075
0.55	0.363
0.575	0.39675
0.6	0.432
0.625	0.46875
0.65	0.507
0.675	0.54675
0.7	0.588
0.725	0.63075
0.75	0.675
0.775	0.72075
0.8	0.768
0.825	0.81675
0.85	0.867
0.875	0.91875
0.9	0.972
0.925	1.02675
0.95	1.083
0.975	1.14075
1	1.2
1.025	1.26075
1.05	1.323
1.075	1.38675
1.1	1.452
1.125	1.51875
1.15	1.587
1.175	1.65675
1.2	1.728
1.225	1.80075
1.25	1.875
1.275	1.95075
1.3	2.028
1.325	2.10675
1.35	2.187
1.375	2.26875
1.4	2.352
1.425	2.43675
1.45	2.523
1.475	2.61075
1.5	2.7
1.525	2.79075
1.55	2.883
1.575	2.97675
1.6	3.072
1.625	3.16875
1.65	3.267
1.675	3.36675
1.7	3.468
1.725	3.57075
1.75	3.675
1.775	3.78075
1.8	3.888
1.825	3.99675
1.85	4.107
1.875	4.21875
1.9	4.332
1.925	4.44675
1.95	4.563
1.975	4.68075
2	4.8

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
Fitted quadratic stiffness	2.4	N/m
Potential offset	1	J
Fitted potential difference at1 m	1.2	J
Withheld true potential difference at2 m	7.2	J
Predicted potential difference at2 m	4.8	J
Withheld energy residual	2.4	J
Independent force at1 m	-2.8	N
Fitted-force prediction at1 m	-2.4	N
Force observed minus predicted	-0.4	N

A chosen potential zero cannot repair a wrong spatial dependence or its force derivative.

From ΔU1=1+q, a purely quadratic fit has stiffness2(1+q). At x2 it predicts4(1+q), while the supplied shape gives4+16q, leaving12q. The independent force packet at x1 differs by−2q N. Both comparisons cancel a common additive energy offset. The force observation is separately supplied, not claimed independent if merely differentiated from the fitted curve.

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

A chosen potential zero cannot repair a wrong spatial dependence or its force derivative.

Observation to attach to a capture

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

Understand

Test a potential shape with evidence not used to fit it

Freeze that fit before revealing U(2)−U(0). The quadratic predicts4(1+q) J, while the supplied shape gives4+16q J. The residual is12q. A common energy offset cancels from both differences and cannot repair the discrepancy.

A separately supplied force packet at x = 1 has Ftrue=−(2+4q) N. Differentiating the fitted quadratic predicts−2(1+q) N, leaving force residual−2q. This is an additional check only because the force value is separately supplied; merely differentiating the already-fitted energy curve would not create new independent evidence.

At q=.2,C = 1, the fitted stiffness is2.4 N/m. The withheld energy difference is7.2 J versus predicted4.8 J, residual2.4 J. The force packet is−2.8 N versus predicted−2.4 N. These observations challenge the quadratic approximation under the disclosed exact assumptions, not uniquely identify a real material’s force law.

Changed-case check

Predict before reading the comparison. At q=.5, withheld energy residual is6 J and force residual−1 N. Changing C from−2 to+2 J changes neither residual nor force.

Retained investigation

Fit only the first difference, predict the withheld energy and independent force, then vary the common offset. Preserve which evidence was used and which was withheld. Why does differentiating a fitted curve not automatically create an independent force measurement?

Source and boundary

Primary conceptual reference checked30 October2026: OpenStax University Physics Volume1. All functions, numerical packets and teaching prose are original. Bounded supplied models are not measured experiments, apparatus instructions, independent practical skill or institutional credit. Numerical algorithm properties are derived for the disclosed model; do not treat them as universal solver guarantees.

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

Why does differentiating a fitted curve not automatically create an independent force measurement?

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