{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# When Models Fail\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/when-models-fail). This notebook includes the lesson text and its Python examples in reading order. Run the setup cell, then run the remaining cells from top to bottom. In Colab, use File → Save a copy in Drive to keep your edits. You can run and edit this view without changing the course source. Examples use illustrative data unless stated otherwise." ] }, { "cell_type": "code", "id": "code-001", "metadata": {}, "source": [ "import importlib.metadata\n", "import subprocess\n", "import sys\n", "\n", "requirements = {\"numpy\": (2, 0), \"sympy\": (1, 13), \"matplotlib\": (3, 8)}\n", "to_install = []\n", "for package, minimum in requirements.items():\n", " try:\n", " installed = importlib.metadata.version(package)\n", " version = tuple(int(part) for part in installed.split(\".\")[:2])\n", " except importlib.metadata.PackageNotFoundError:\n", " version = (0, 0)\n", " if version < minimum:\n", " to_install.append(f\"{package}>={minimum[0]}.{minimum[1]}\")\n", "if to_install:\n", " subprocess.check_call([sys.executable, \"-m\", \"pip\", \"install\", *to_install])\n", "\n", "import numpy as np\n", "import sympy as sp\n", "import matplotlib\n", "import matplotlib.pyplot as plt\n", "\n", "print(\"Python\", sys.version.split()[0])\n", "print(\"NumPy\", np.__version__)\n", "print(\"SymPy\", sp.__version__)\n", "print(\"Matplotlib\", matplotlib.__version__)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-002", "metadata": {}, "source": [ "A mismatch between a prediction and reality does not have one universal cause. The\n", "equation may have been copied incorrectly, the code may solve a different equation,\n", "the numerical method may be inaccurate or unstable, an input may have the wrong\n", "unit, the measurements may be biased, or the physical approximation may omit an\n", "important interaction. Scientific debugging means distinguishing these cases\n", "rather than adjusting parameters until a graph looks convincing.\n", "\n", "Failure is useful evidence when it is preserved and classified. It tells us which\n", "claim did not survive a test and what kind of refinement is needed. Hiding a\n", "failure with smoothing, an undocumented correction factor, or a wider plotting\n", "scale discards the most informative part of the investigation.\n", "\n", "## Define what “failure” means\n", "\n", "A model cannot pass or fail without a stated purpose and tolerance. An error of one\n", "millimetre may be irrelevant in an astronomical estimate and unacceptable in a\n", "precision instrument. Before comparing values, specify:\n", "\n", "- the observable being compared;\n", "- the same units, coordinate system, and reference frame for both values;\n", "- measurement uncertainty and numerical tolerance;\n", "- the spatial, temporal, or parameter range;\n", "- the criterion that would count as adequate agreement.\n", "\n", "For a reference value $q_{\\mathrm{ref}}$\n", "and a prediction $q_{\\mathrm{model}}$,\n", "the absolute discrepancy is\n", "\n", "\n", "$$\n", "\\Delta q=q_{\\mathrm{model}}-q_{\\mathrm{ref}}\n", "$$\n", "\n", "\n", "and, when the reference is not zero, a useful relative measure is\n", "\n", "\n", "$$\n", "\\epsilon_{\\mathrm{rel}}=\\frac{|q_{\\mathrm{model}}-q_{\\mathrm{ref}}|}{|q_{\\mathrm{ref}}|}\n", "$$\n", "\n", "\n", "Relative error is undefined when the denominator is zero and can be misleading\n", "when it is merely close to zero. In those cases, use an absolute tolerance tied to\n", "the physical scale or combine absolute and relative criteria." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "prediction, reference = 9.9, 10.0\n", "absolute_error = abs(prediction - reference)\n", "allowed = 0.01 + 0.02 * abs(reference)\n", "print(\"Within the declared error allowance:\", absolute_error <= allowed)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The tolerance must be chosen before seeing the answer when possible. Moving it\n", "afterward to make a result pass is not validation.\n", "\n", "## First check the question and definitions\n", "\n", "Many apparent physics failures are comparison failures. One source may report\n", "speed while another reports one component of velocity. One energy value may\n", "include rest energy while another reports only kinetic energy. A thermodynamics\n", "reference may define work done by the gas as positive while an implementation\n", "defines work done on the gas as positive.\n", "\n", "Rewrite both results using the same definitions before diagnosing deeper causes.\n", "Check the system boundary, sign convention, coordinate origin, reference frame,\n", "zero of potential energy, and time at which the quantity is evaluated. A model\n", "cannot agree with a reference answer to a different problem.\n", "\n", "## Check algebra and dimensions\n", "\n", "Return to the governing relationship before inspecting code. Confirm every\n", "rearrangement, exponent, sign, vector direction, and constant. Then check\n", "dimensions. For example, in\n", "\n", "\n", "$$\n", "x(t)=x_0+v_0t+\\frac{1}{2}at^2\n", "$$\n", "\n", "\n", "all terms have dimensions of length. If a term instead contains\n", "$at$, it has dimensions of\n", "speed and cannot be added to position. Dimensional consistency does not prove the\n", "formula, but dimensional inconsistency proves something is wrong.\n", "\n", "Order-of-magnitude reasoning is another early filter. If a walking-speed input\n", "produces an interplanetary distance in one second, inspect powers of ten and unit\n", "conversion before debating subtle physical effects.\n", "\n", "## Verify the implementation\n", "\n", "Verification tests whether the program follows the stated mathematical model.\n", "Useful tests isolate small pieces: a known analytic solution, a zero-input case, a\n", "symmetry, a derivative checked by finite differences, or a conserved quantity in\n", "a system where conservation applies." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "def fall_distance(steps):\n", " distance, velocity, dt = 0.0, 0.0, 1.0 / steps\n", " for _ in range(steps):\n", " distance += velocity * dt\n", " velocity += 9.81 * dt\n", " return distance\n", "\n", "\n", "expected = 0.5 * 9.81\n", "computed = fall_distance(1000)\n", "assert abs(computed - expected) < 0.005\n", "print(\"Known-case absolute error (m):\", abs(computed - expected))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "A passing unit test proves only the tested behaviour. It does not establish that\n", "the mathematical model is appropriate for a new regime. Tests should include\n", "boundaries and invalid inputs, not only a convenient central example.\n", "\n", "Common implementation failures include updating state in the wrong order, mutating\n", "an input that should remain fixed, mixing degrees and radians, using integer\n", "division, losing a minus sign, comparing floating-point values for exact equality,\n", "and reading rounded display output back into a calculation.\n", "\n", "## Separate numerical error from physical error\n", "\n", "A numerical method replaces continuous mathematics with finite steps, samples, or\n", "precision. Its result can fail even when the differential equation is correct. If\n", "a method has error proportional to a power of the step size,\n", "\n", "\n", "$$\n", "E(h)\\approx C h^p\n", "$$\n", "\n", "\n", "then $h$ is the resolution,\n", "$p$ is the method's order, and\n", "$C$ depends on the\n", "problem. Halving the step should reduce the leading error by approximately\n", "$2^p$\n", "once the calculation is in the asymptotic convergence range." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "coarse, medium, fine = [fall_distance(n) for n in [100, 200, 400]]\n", "coarse_change, fine_change = abs(coarse - medium), abs(medium - fine)\n", "assert fine_change < coarse_change\n", "print(\"Successive differences (m):\", coarse_change, fine_change)\n", "print(\"This convergence test does not establish that air drag is negligible.\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Three resolutions reveal more than two because they show whether changes are\n", "decreasing systematically. Convergence toward a stable number does not guarantee\n", "physical truth: the program may be converging accurately to the solution of an\n", "inappropriate equation.\n", "\n", "## Instability is not new physics\n", "\n", "Some algorithms magnify small numerical errors. An explicit update may oscillate\n", "or grow without bound when its step exceeds a stability limit. A visually dramatic\n", "result is then a property of the discretization, not evidence of an explosive\n", "physical process.\n", "\n", "Stability limits depend on the equation and method. They must be derived or taken\n", "from a trustworthy numerical analysis, expressed with consistent units, and\n", "checked before calculation. Reducing a step until a plot looks smooth is not\n", "enough; demonstrate convergence of quantities that matter.\n", "\n", "Round-off can dominate at the opposite extreme. Subtracting nearly equal large\n", "numbers loses significant digits, and taking excessively small finite-difference\n", "steps can make round-off worse even while truncation error improves. Resolution\n", "studies should therefore look for a range in which the result stabilizes, not\n", "assume smaller is always better.\n", "\n", "## Inspect inputs and data provenance\n", "\n", "Correct code with correct equations still fails when its inputs do not represent\n", "the intended system. Check units at the data boundary, missing-value handling,\n", "timestamps, sensor calibration, coordinate transformations, sample selection, and\n", "whether constants match the reference version.\n", "\n", "Measured data contain uncertainty. If the model-reference discrepancy is smaller\n", "than the combined uncertainty, the data may not distinguish the model from the\n", "reference. If residuals show a systematic pattern rather than random scatter, a\n", "missing mechanism or calibration problem may be present.\n", "\n", "For observations $y_i$ and\n", "predictions $f_i$, residuals\n", "are\n", "\n", "\n", "$$\n", "r_i=y_i-f_i\n", "$$\n", "\n", "\n", "A single summary number can hide structure. Plot or tabulate residuals against\n", "time, position, and relevant inputs. A curve in residuals, a change in spread, or\n", "clusters by instrument may identify the source of failure.\n", "\n", "## Decide whether the physical approximation has failed\n", "\n", "Only after verifying definitions, equations, code, numerics, and inputs should we\n", "attribute the mismatch to the physical model. Then identify a specific assumption:\n", "\n", "- a projectile may require drag or a changing gravitational field;\n", "- a collision may exchange momentum with an environment outside the chosen system;\n", "- an ideal gas may be too dense or cold for negligible molecular interactions;\n", "- a circuit element may heat enough that its resistance is no longer constant;\n", "- a nonrelativistic formula may be used at speeds where relativistic effects matter;\n", "- a point-particle approximation may fail when size, rotation, or deformation matters.\n", "\n", "The failure boundary is often gradual. Compare the neglected effect with the\n", "retained one using a dimensionless ratio. A correction that is one part in a\n", "million may be irrelevant for a classroom estimate and essential for a precision\n", "measurement.\n", "\n", "## Avoid overfitting the failure\n", "\n", "Adding parameters can make a model follow existing data more closely without\n", "improving prediction. A flexible model may fit measurement noise. Preserve data\n", "that were not used to choose parameters and evaluate the revised model on that\n", "independent set. Prefer a physically motivated correction over an unexplained\n", "factor inserted only to erase residuals.\n", "\n", "If several models fit within measurement uncertainty, report that the evidence\n", "does not distinguish them. Scientific honesty permits an inconclusive result.\n", "Selecting the most complicated explanation is not progress when the data cannot\n", "test its extra claims.\n", "\n", "## A disciplined failure report\n", "\n", "A useful report makes the investigation reproducible:\n", "\n", "1. State the expected and observed quantities with units and uncertainty.\n", "2. Give the signed discrepancy and the predeclared acceptance criterion.\n", "3. Confirm aligned definitions, frames, conventions, and system boundaries.\n", "4. List implementation tests and their results.\n", "5. Show numerical convergence or explain why it is unavailable.\n", "6. Record data sources, versions, transformations, and exclusions.\n", "7. Identify the assumption most directly challenged by the remaining evidence.\n", "8. Propose a focused next test that could distinguish competing explanations.\n", "\n", "Do not silently delete failed runs. Store enough configuration, software version,\n", "random seed, and diagnostic output to reproduce them. A clear failure report is\n", "more valuable than a polished success whose assumptions cannot be recovered.\n", "\n", "The aim is not to make every model agree forever. A model earns trust by succeeding\n", "inside a documented domain, failing visibly outside it, and making the reason for\n", "revision inspectable. That cycle is how simplified laws become reliable tools\n", "without being mistaken for complete descriptions of nature.\n", "\n", "## A worked Python notebook\n", "\n", "Quantify the departure from uniform gravity as height becomes a significant fraction of Earth's radius. The question for this calculation is precise: how does **inverse-square gravity relative to surface gravity**\n", "change when we change **height above Earth surface**, while keeping the other choices fixed?\n", "The earlier sections explain the physical relationship. Here we turn one part of that\n", "relationship into a calculation that you can run, inspect, and change in Google Colab.\n", "\n", "Earth is treated as spherical; this isolates height dependence from rotation and local mass variation. Keep that sentence beside the results. A formula can return a number outside\n", "its useful range, and a plotting library will usually draw that number without asking\n", "whether the physical assumptions still hold. The selected interval is from\n", "0 to 1e+06 m. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for height above Earth surface, expressed in m. The output is\n", "inverse-square gravity relative to surface gravity, expressed in dimensionless. The equation below uses the numerical values in those\n", "units. Any fixed coefficients therefore carry the units needed to make the relation\n", "consistent. A value supplied in another unit must be converted before it enters this\n", "expression.\n", "\n", "\n", "$$\n", "f(q) = \\frac{40589641000000.0}{\\left(q + 6371000.0\\right)^{2}}\n", "$$\n", "\n", "\n", "This is a function of one changing input. It is not a statement that the physical\n", "system has only one variable. We deliberately hold the other inputs fixed so that one\n", "cause of variation is visible. Changing several inputs at once would make it harder\n", "to tell which change produced the output difference. Write down the held-fixed choices\n", "before trying a different experiment.\n", "\n", "At 100000 m, the expression gives 0.969332 dimensionless. This is our\n", "reference calculation. It is a prediction of the stated example, not a new measurement.\n", "Its displayed digits help us compare implementations. They should not be read as a\n", "claim that physical inputs are known to six significant figures." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "import numpy as np\n", "import sympy as sp\n", "import matplotlib.pyplot as plt\n", "\n", "q = sp.Symbol(\"q\", real=True)\n", "expression = sp.sympify(\"(6.371e6/(6.371e6+q))**2\", locals={\"q\": q})\n", "reference_input = 100000.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\n", " \"Predicted inverse-square gravity relative to surface gravity:\",\n", " reference_output,\n", " \"dimensionless\",\n", ")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "1 dimensionless. The midpoint, at 500000 m, gives\n", "0.859756 dimensionless. The upper endpoint gives 0.747072 dimensionless.\n", "Read these together with the assumptions: they describe different choices of height above Earth surface\n", "within the same example, with the other inputs held fixed.\n", "\n", "These three values are a useful first check, but they do not establish the shape between\n", "them. Two endpoints can be equal even when the curve bends or oscillates in between.\n", "Likewise, a large change between endpoints does not tell us whether the relationship is\n", "linear. We need the equation and intermediate samples to answer those questions." ] }, { "cell_type": "code", "id": "code-011", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.0, 500000.0, 1000000.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} m -> {output_value:.6g} dimensionless\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "metadata": {}, "source": [ "`lambdify` turns the symbolic expression into a numerical function. We explicitly select\n", "NumPy as its numerical backend so it can accept an array. The same expression now has\n", "two useful forms: one for algebra and one for calculating many cases. The assertion\n", "checks that the chosen inputs produce finite outputs. It would catch an infinity or an\n", "undefined numerical result, but a finite value can still be physically inappropriate.\n", "\n", "### Make a graph that answers the question\n", "\n", "The horizontal axis shows height above Earth surface; the vertical axis shows inverse-square gravity relative to surface gravity. Every plotted\n", "point is a prediction of the same expression. We mark the reference case separately\n", "so that a reader can connect the earlier printed value to its place on the curve.\n", "The axis labels include units because a shape without a scale cannot tell us the size\n", "of a physical effect." ] }, { "cell_type": "code", "id": "code-013", "metadata": {}, "source": [ "input_grid = np.linspace(0.0, 1000000.0, 301)\n", "output_grid = np.asarray(evaluate(input_grid), dtype=float)\n", "output_grid = np.broadcast_to(output_grid, input_grid.shape)\n", "fig, ax = plt.subplots(figsize=(7, 4))\n", "ax.plot(input_grid, output_grid, label=\"Stated model\")\n", "ax.scatter([reference_input], [reference_output], label=\"Reference case\", zorder=3)\n", "ax.set_xlabel(\"Height above earth surface (m)\")\n", "ax.set_ylabel(\"Inverse-square gravity relative to surface gravity (dimensionless)\")\n", "ax.legend()\n", "ax.grid(alpha=0.25)\n", "fig.tight_layout()\n", "plt.show()\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "metadata": {}, "source": [ "Look for a sign change, a turning point, or a region where the curve becomes steep.\n", "Connect each feature to the expression rather than to the colour of the line. If the\n", "curve has a minimum or maximum, changing the input in opposite directions may have\n", "similar effects on the output near that point. If it is nearly straight over a short\n", "interval, a local linear approximation may be useful even when the complete expression\n", "is nonlinear.\n", "\n", "Increasing the number of samples makes the line smoother and may reveal features that\n", "a coarse grid missed. It does not improve the physical inputs or enlarge the model's\n", "validity range. For quantities that can take only discrete values, use the line as a\n", "guide to the relationship and interpret only the allowed input values. A plotting\n", "function does not enforce such restrictions for you.\n", "\n", "### Measure the local sensitivity\n", "\n", "A derivative asks a more specific question than the whole graph: how quickly does\n", "inverse-square gravity relative to surface gravity change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 100000 m is\n", "-2.99593e-07 in those ratio units. Locally, the output decreases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = - \\frac{81179282000000.0}{\\left(q + 6371000.0\\right)^{3}}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 105000.0\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"dimensionless\")\n", "print(\"Full expression:\", full_prediction, \"dimensionless\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"dimensionless\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 105000 m. The tangent estimate is\n", "0.967834 dimensionless, while the full expression gives\n", "0.967835 dimensionless. If the two values are close, it means the chosen\n", "input change is small enough for this local approximation at this point. It does not\n", "mean the entire model is linear. If the slope is zero, a quadratic or higher-order term\n", "may supply the leading change instead.\n", "\n", "This is also the beginning of uncertainty propagation. If the input has a small stated\n", "standard uncertainty, the magnitude of the slope times that uncertainty estimates one\n", "contribution to output uncertainty. Do not automatically use the same rule for a wide\n", "interval, a discontinuity, a turning point with zero first derivative, or several\n", "correlated uncertain inputs. Those situations need more of the model than one slope.\n", "\n", "### Check the derivative with numerical differences\n", "\n", "We can estimate the same slope from values on either side of the reference point.\n", "Subtract the lower output from the upper output and divide by twice the input step.\n", "Parentheses are essential: the entire output difference must be divided by the entire\n", "input interval. Without them, Python can execute a different mathematical expression\n", "while still producing a plausible number.\n", "\n", "The calculation below uses three decreasing step sizes. Each estimate is compared with\n", "the symbolic derivative, so the reference is independent of the numerical difference\n", "formula. Near an endpoint, it is acceptable here to evaluate a small extension only\n", "because the displayed mathematical expression is smooth there; such a check does not\n", "assert that extrapolated values describe additional physical cases." ] }, { "cell_type": "code", "id": "code-017", "metadata": {}, "source": [ "input_span = 1000000.0\n", "for fraction in [1e-2, 1e-3, 1e-4]:\n", " step = input_span * fraction\n", " upper = float(evaluate(reference_input + step))\n", " lower = float(evaluate(reference_input - step))\n", " numerical_slope = (upper - lower) / (2 * step)\n", " difference = numerical_slope - slope_at_reference\n", " print(f\"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "For a smooth function, reducing the step often reduces the approximation error at first.\n", "Very small steps can eventually make subtraction of nearly equal floating-point values\n", "less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare\n", "the observed behaviour over a useful range of steps instead of choosing the smallest\n", "number the computer can represent.\n", "\n", "An absolute slope difference is useful when the expected slope is near zero. A relative\n", "difference would divide by a very small reference and could look enormous even when the\n", "absolute discrepancy is tiny. Away from zero, a relative difference can help compare\n", "quantities with very different scales. The tolerance should reflect the question and\n", "the numerical method, rather than a universal decimal threshold copied into every test.\n", "\n", "### Change the example and preserve the reasoning\n", "\n", "Make one copy of the reference calculation before editing it. Choose a second value of\n", "height above Earth surface inside the stated interval, predict whether inverse-square gravity relative to surface gravity will rise or fall,\n", "and then evaluate the expression. Place the prediction and the result in a text cell.\n", "If they disagree, identify whether the mistake was in the sign, the unit conversion,\n", "the algebra, or the assumption about how the curve behaves.\n", "\n", "Next choose a shorter plotting interval around the reference point. Compare the local\n", "curve with its tangent approximation. Then widen the interval again. The difference\n", "shows why a useful local estimate need not describe the whole range. Keep the axis\n", "labels and numerical scale visible when comparing the plots; automatic axis limits can\n", "make two different effects appear equally large.\n", "\n", "Finally, return to the physical limit stated at the start: Earth is treated as spherical; this isolates height dependence from rotation and local mass variation. Describe one change\n", "to the setup that would violate that limit. State what new input, equation, or observation\n", "would be needed before trusting an answer in the changed situation. This last step\n", "connects the Python experiment to the physics instead of treating successful execution\n", "as the only criterion for a useful result.\n", "\n", "Before saving, restart the runtime and run all cells in order. A result that depends on\n", "an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep\n", "unrounded values for later calculations, round only the reported output, and label any\n", "new data as measured, assumed, or generated. The graph and printed values should remain\n", "traceable to the same expression and the same choices of inputs.\n", "\n", "### Recover the output change by integrating the slope\n", "\n", "There is another way to check that the expression and its derivative describe the same\n", "relationship. Add up the small output changes predicted by the slope across the chosen\n", "input interval. In the continuous limit, this integral equals the output at the upper\n", "endpoint minus the output at the lower endpoint. The result has units of dimensionless:\n", "the input unit in the integration cancels the input unit in the denominator of the slope.\n", "\n", "This statement is about the mathematics of this particular relationship. It does not\n", "assign a new physical meaning to an arbitrary area under a graph. For example, an area\n", "under force versus distance represents work, while an area under a different pair of\n", "quantities may represent something else or have no standard name. Always carry the\n", "axis units through the calculation before naming an integral." ] }, { "cell_type": "code", "id": "code-019", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(1000000.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 1000000.0, points)\n", " slopes = np.asarray(slope_function(grid), dtype=float)\n", " slopes = np.broadcast_to(slopes, grid.shape)\n", " integrated_change = np.trapezoid(slopes, grid)\n", " print(\n", " points,\n", " \"integrated change:\",\n", " integrated_change,\n", " \"endpoint change:\",\n", " endpoint_change,\n", " )\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-020", "metadata": {}, "source": [ "The trapezoidal rule joins neighbouring slope samples with straight segments and adds\n", "their signed areas. A negative contribution remains negative; replacing every slope\n", "by its magnitude would calculate a different quantity. Increasing the sample count\n", "provides a practical refinement check. If the values do not settle toward the endpoint\n", "difference, first inspect the interval, singularities, and array ordering.\n", "\n", "Do not conclude that two agreeing calculations prove the model is physically complete.\n", "Both use the same selected relationship, so an omitted effect would be omitted from\n", "both. Their agreement is evidence that the numerical tools implement that relationship\n", "consistently. Comparing with an independent experiment or a more appropriate physical\n", "model is a separate test.\n", "\n", "Keep this comparison in the notebook when you change the expression. It is particularly\n", "useful after adding a coefficient, changing a sign convention, or converting an input\n", "unit. A code cell that produces a smooth graph can conceal these mistakes; a second\n", "calculation of the same output change gives you another place to notice them.\n", "\n", "## Sources\n", "\n", "- [OpenStax, *University Physics Volume 1*, Section 1.4: Approximation](https://openstax.org/books/university-physics-volume-1/pages/1-4-approximation).\n", "- [OpenStax, *University Physics Volume 1*, Section 3.4: Motion with Constant Acceleration](https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration).\n", "- [MIT OpenCourseWare, 8.01SC Classical Mechanics: Online Textbook](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/pages/online-textbook/).\n", "- [The Physics Hypertextbook](https://physics.info/).\n", "- [Physics LibreTexts, Numerical Methods and Physics](https://phys.libretexts.org/Bookshelves/Mathematical_Physics_and_Pedagogy).\n", "- Benjamin Crowell, *Light and Matter*, locally archived course reference." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "when-models-fail.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/when-models-fail", "source_sha256": "7cb476fddadc8a33a1c7d308ae278a6be04fef476d5d02bb335bad1283b3ab12" } }, "nbformat": 4, "nbformat_minor": 5 }