SCIENCE · INVESTIGATE AND EXPLAIN

Normalize speed evidence without hiding the omitted tail

The Maxwell speed density describes nonnegative speeds, not signed velocity components. Write u=v/s with s=sqrt(2kT/m). Then f_v(v)dv=(4/sqrt(pi))u² exp(−u²)du. The factor u² counts velocity-space directions at a given speed; omitting it would produce a different probability distribution.

Learning outcome: Derive the first two moments, retain cutoff and numerical-resolution roles separately, and explain why a normalized density need not be a validated physical model.

Starting recap

Review definite integrals, probability densities, mean versus mean square, temperature units and the first-law work convention. State the domain and distinguish calibration data from unused validation.

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Try

At s=200 m/s, mode is 200 m/s, mean approximately 225.675833 m/s and RMS approximately 244.948974 m/s. At L=3, the tail upper bound is exp(−9)(6+1/3)/sqrt(pi), approximately 0.000441. A cutoff at 3s excludes a small but nonzero probability.

Open the experiment

Predict

Which statement correctly separates the displayed error controls?

Your prediction and reason

Experiment

Derive the first two moments, retain cutoff and numerical-resolution roles separately, and explain why a normalized density need not be a validated physical model.

Compare a finite integral with an infinite distribution

Worked teaching model · synthetic values · Maxwell speed distribution with u=v/s and density 4u² exp(−u²)/sqrt(pi). Scale s=sqrt(2kT/m). Simpson sums use 160 and 320 panels on 0..L; their signed difference is a diagnostic, not a rigorous error bound; an analytic upper bound limits the omitted tail, not numerical quadrature error. Ideal dilute classical equilibrium only.

Thermal speed scale: 200 m/s Integration cutoff divided by speed scale: 3 ratio

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Dimensionless Maxwell speed density

Graph axes rescale independently; read units and values, not screen angles. 1: normalized infinite-domain model (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Dimensionless Maxwell speed density

normalized infinite-domain model

u = speed/scale	density per unit u
0	0
0.0625	0.008781094
0.125	0.03471516
0.1875	0.07659835
0.25	0.1325018
0.3125	0.1998819
0.375	0.2757242
0.4375	0.3567093
0.5	0.4393913
0.5625	0.5203752
0.625	0.596484
0.6875	0.6649033
0.75	0.7232975
0.8125	0.7698916
0.875	0.8035159
0.9375	0.8236135
1	0.830215
1.0625	0.8238823
1.125	0.805631
1.1875	0.7768358
1.25	0.7391285
1.3125	0.6942943
1.375	0.6441729
1.4375	0.5905707
1.5	0.5351863
1.5625	0.4795522
1.625	0.4249941
1.6875	0.3726072
1.75	0.3232469
1.8125	0.2775338
1.875	0.2358691
1.9375	0.1984576
2	0.1653359
2.0625	0.1364033
2.125	0.111453
2.1875	0.0902016
2.25	0.07231598
2.3125	0.05743689
2.375	0.04519808
2.4375	0.0352415
2.5	0.02722855
2.5625	0.02084773
2.625	0.01581915
2.6875	0.01189658
2.75	0.008867443
2.8125	0.006551382
2.875	0.004797836
2.9375	0.003483008
3	0.002506555
3.0625	0.001788257
3.125	0.001264818
3.1875	0.0008869224
3.25	0.0006166182
3.3125	0.0004250429
3.375	0.0002905006
3.4375	0.0001968659
3.5	0.000132286
3.5625	0.00008814285
3.625	0.00005823715
3.6875	0.00003815593
3.75	0.00002479028
3.8125	0.00001597227
3.875	0.00001020529
3.9375	0.00000646644
4	0.000004063435
4.0625	0.000002532307
4.125	0.000001565096
4.1875	9.593412e-7
4.25	5.832018e-7
4.3125	3.516283e-7
4.375	2.102681e-7
4.4375	1.247075e-7
4.5	7.335789e-8
4.5625	4.279967e-8
4.625	2.476727e-8
4.6875	1.421557e-8
4.75	8.092879e-9
4.8125	4.569814e-9
4.875	2.559496e-9
4.9375	1.421919e-9
5	7.835433e-10

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
Finite-cutoff Simpson normalization	0.9995602	ratio
Coarser 160-panel normalization	0.9995602	ratio
Signed 320-minus-160 resolution difference	1.870789e-10	ratio
Analytic omitted-probability upper bound	0.000440968	ratio
Most probable speed	200	m/s
Mean speed	225.6758	m/s
Root-mean-square speed	244.949	m/s
Finite integral of u squared density	1.49558	ratio
Infinite-domain second moment of u	1.5	ratio

The distribution is normalized over infinity; a finite integral and its tail need separate accounting.

Substitute u=v/s so f_v(v)dv=(4/sqrt(pi))u²e^(−u²)du. Gaussian moments give mode s, mean 2s/sqrt(pi) and RMS sqrt(3/2)s. Integration by parts gives tail=(2L/sqrt(pi))e^(−L²)+erfc(L). Bounding the Gaussian remainder by e^(−L²)/(sqrt(pi)L) gives the displayed upper bound. This analytic truncation bound is not a certified Simpson discretization bound; compare resolutions independently.

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

The distribution is normalized over infinity; a finite integral and its tail need separate accounting.

Observation to attach to a capture

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

Understand

Declare the distribution or system

Derive the integral or response

Normalization over 0..infinity follows from the Gaussian moment integral. Differentiating the density gives its nonzero maximum at u=1. Its first moment is 2/sqrt(pi), while its second moment is 3/2. Thus mode s, mean 2s/sqrt(pi) and RMS sqrt(3/2)s are distinct. A probability density has units inverse speed; its height at one speed is not the probability of exactly that speed.

Audit limits and evidence

The interactive integral stops at u=L and displays Simpson sums with 160 and 320 panels plus their signed difference. That difference is a resolution diagnostic, not a rigorous error bound. Integration by parts gives omitted probability (2L/sqrt(pi))exp(−L²)+erfc(L). For L>0, bounding the remaining Gaussian integral gives erfc(L)≤exp(−L²)/(sqrt(pi)L), yielding the displayed rigorous tail bound. That bound controls omitted probability only, not quadrature error or the omitted second moment.

Predict a changed premise

Compare two cutoffs and independently compare two integration resolutions. A finite integral close to one can support numerical consistency without establishing the physical Maxwell premise. A directed beam, strong nonequilibrium drive or quantum-degenerate gas would require another distribution even if this integral were evaluated exactly.

Worked original case

Investigation

Derive the first two moments, retain cutoff and numerical-resolution roles separately, and explain why a normalized density need not be a validated physical model.

Changed-case practice

Try the changed case before feedback

For speed scale s=300 m/s, find the RMS speed in m/s.

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.

Sources and boundaries

Original derivations and synthetic cases checked 31 October 2026 against the primary teaching reference. No source exercises or graphics are reproduced. These bounded CPU calculations do not claim actual gas handling, pressure-vessel work, thermal exposure or practical measurement. Public feedback is practice, not independently verified reasoning; retain original work, critique, revision and a newly assigned case in the course evidence workspace.

The two bounded Simpson sums use the rule explained in OpenStax Calculus II numerical integration; the signed resolution difference is not its rigorous derivative-based error bound.

Apply

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

Which distribution, constitutive law, path or calibration premise supports this inference, and what independent evidence would challenge it?

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: Integrate a velocity distribution into directional pressure Next: Derive the speed bias of a flux-selected sample

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