Separate a physical time constant from a displayed one
The time coordinate itself can require calibration. Suppose displayed time s equals g times physical time t, with exact zero but unknown positive gain g. A zero-offset voltage 4e^(−t/τ) becomes 4e^[−s/(gτ)] on the displayed axis. The product gτ, not either factor separately, controls every point on that graph.
Learning outcome: Find two generator pairs with the same product gτ and compare their entire displayed curves. Then apply the independent calibration interval and state whether its premise covers each selected gain. Keep response validation distinct from clock validation.
Starting recap
Review charge continuity, definite integrals, first-order differential equations, initial conditions and signed source work. Distinguish displayed time from physical time.
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Try
For physical τ=1 s and clock gain 1.1, the displayed time constant is 1.1 display s. A second pair τ=1.1 s and g=1 gives the identical displayed curve. Applying the proposed narrow calibration would give approximately 1.078431 to 1.122449 s, which excludes the first true τ because its gain violates the premise.
Open the experimentPredict
What does the displayed voltage response identify without independent clock evidence?
Experiment
Find two generator pairs with the same product gτ and compare their entire displayed curves. Then apply the independent calibration interval and state whether its premise covers each selected gain. Keep response validation distinct from clock validation.
Separate a displayed timescale from a physical one
Worked teaching model · synthetic values · Display time s=g t, with exact zero and positive clock gain g. Zero-offset voltage is 4 exp(−t/tau). Two records at displayed times 0 and 1 s fit only g tau. A fresh displayed-time record at 2 s tests that shape but cannot identify g separately. A distinct proposed clock calibration g in [0.98,1.02] is conditional, not asserted true for every slider setting.
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A clock-scaled response family
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: fit identifies only g times tau (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for A clock-scaled response family
fit identifies only g times tau
| displayed time (display s) | voltage (V) |
|---|---|
| 0 | 4 |
| 0.0375 | 3.865935 |
| 0.075 | 3.736362 |
| 0.1125 | 3.611133 |
| 0.15 | 3.490101 |
| 0.1875 | 3.373126 |
| 0.225 | 3.260071 |
| 0.2625 | 3.150805 |
| 0.3 | 3.045202 |
| 0.3375 | 2.943137 |
| 0.375 | 2.844494 |
| 0.4125 | 2.749157 |
| 0.45 | 2.657015 |
| 0.4875 | 2.567962 |
| 0.525 | 2.481893 |
| 0.5625 | 2.398709 |
| 0.6 | 2.318313 |
| 0.6375 | 2.240612 |
| 0.675 | 2.165515 |
| 0.7125 | 2.092934 |
| 0.75 | 2.022787 |
| 0.7875 | 1.95499 |
| 0.825 | 1.889466 |
| 0.8625 | 1.826138 |
| 0.9 | 1.764933 |
| 0.9375 | 1.705779 |
| 0.975 | 1.648607 |
| 1.0125 | 1.593352 |
| 1.05 | 1.539948 |
| 1.0875 | 1.488335 |
| 1.125 | 1.438451 |
| 1.1625 | 1.39024 |
| 1.2 | 1.343644 |
| 1.2375 | 1.29861 |
| 1.275 | 1.255085 |
| 1.3125 | 1.213019 |
| 1.35 | 1.172363 |
| 1.3875 | 1.13307 |
| 1.425 | 1.095094 |
| 1.4625 | 1.05839 |
| 1.5 | 1.022917 |
| 1.5375 | 0.9886322 |
| 1.575 | 0.9554968 |
| 1.6125 | 0.9234721 |
| 1.65 | 0.8925206 |
| 1.6875 | 0.8626066 |
| 1.725 | 0.8336952 |
| 1.7625 | 0.8057527 |
| 1.8 | 0.7787468 |
| 1.8375 | 0.7526461 |
| 1.875 | 0.7274201 |
| 1.9125 | 0.7030396 |
| 1.95 | 0.6794763 |
| 1.9875 | 0.6567027 |
| 2.025 | 0.6346924 |
| 2.0625 | 0.6134199 |
| 2.1 | 0.5928603 |
| 2.1375 | 0.5729897 |
| 2.175 | 0.5537852 |
| 2.2125 | 0.5352243 |
| 2.25 | 0.5172856 |
| 2.2875 | 0.499948 |
| 2.325 | 0.4831916 |
| 2.3625 | 0.4669968 |
| 2.4 | 0.4513447 |
| 2.4375 | 0.4362173 |
| 2.475 | 0.4215969 |
| 2.5125 | 0.4074665 |
| 2.55 | 0.3938097 |
| 2.5875 | 0.3806106 |
| 2.625 | 0.367854 |
| 2.6625 | 0.3555248 |
| 2.7 | 0.3436089 |
| 2.7375 | 0.3320924 |
| 2.775 | 0.3209619 |
| 2.8125 | 0.3102044 |
| 2.85 | 0.2998075 |
| 2.8875 | 0.289759 |
| 2.925 | 0.2800474 |
| 2.9625 | 0.2706612 |
| 3 | 0.2615896 |
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.
| Quantity | Value | Unit |
|---|---|---|
| Reading at displayed one second | 1.611561 | V |
| Fitted displayed time constant | 1.1 | display s |
| Fresh displayed two-second prediction | 0.6492824 | V |
| Fresh displayed two-second observation | 0.6492824 | V |
| Conditional physical time lower endpoint | 1.078431 | s |
| Conditional physical time upper endpoint | 1.122449 | s |
| Generator physical time constant | 1 | s |
| Generator clock gain | 1.1 | ratio |
A fresh voltage sample tests the displayed response without independently calibrating the clock.
Because displayed s=g t, the response is 4 exp[−s/(g tau)]. Every positive pair with the same product g tau gives the same voltage versus displayed time. Only an independent clock constraint can bound physical tau. The proposed calibration [0.98,1.02] gives fit/1.02 through fit/0.98; if the selected generator gain lies outside that interval, the premise is false and the interval need not contain the true tau.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
A fresh voltage sample tests the displayed response without independently calibrating the clock.
Because displayed s=g t, the response is 4 exp[−s/(g tau)]. Every positive pair with the same product g tau gives the same voltage versus displayed time. Only an independent clock constraint can bound physical tau. The proposed calibration [0.98,1.02] gives fit/1.02 through fit/0.98; if the selected generator gain lies outside that interval, the premise is false and the interval need not contain the true tau.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
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Which initial state, calibration or constitutive assumption is needed for this conclusion, and what fresh evidence could test it?
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