{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Why code makes physics easier\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/why-physics-needs-code). 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": [ "Physics is not a collection of formulas to memorize. It is a way to build models,\n", "derive predictions from them, and compare those predictions with observations. Code is\n", "useful because it makes that whole chain explicit. It can repeat a calculation, vary an\n", "input, preserve units in variable names, and turn a physical claim into a test. Code does\n", "not make a model true, but it makes the consequences of the model easier to inspect.\n", "\n", "## From a question to a model\n", "\n", "Suppose you release a small dense object from a height near Earth's surface and want to\n", "estimate when it reaches the ground. Before writing code, decide what the simplified\n", "world contains. Treat the object as a particle, choose downward as the positive\n", "direction, neglect air resistance, assume it starts from rest, and use a constant\n", "gravitational acceleration.\n", "\n", "Under those assumptions, the downward distance after elapsed time\n", "$t$ is\n", "\n", "\n", "$$\n", "d(t)=\\frac{1}{2}gt^2\n", "$$\n", "\n", "\n", "Here $d$ is measured in metres,\n", "$g$ in metres per second\n", "squared, and $t$ in seconds. The units\n", "check before any numbers are substituted:\n", "\n", "\n", "$$\n", "[d]=[g][t]^2=\\frac{\\mathrm{m}}{\\mathrm{s}^2}\\,\\mathrm{s}^2=\\mathrm{m}\n", "$$\n", "\n", "\n", "The notation $[d]$ means \"the\n", "dimensions of $d$.\" A dimensional match\n", "does not prove the equation, but a mismatch proves that something is wrong.\n", "\n", "If the release height is $h$, landing\n", "occurs when the distance fallen equals that height. Solving the same relationship for\n", "time gives\n", "\n", "\n", "$$\n", "t_{\\mathrm{land}}=\\sqrt{\\frac{2h}{g}}\n", "$$\n", "\n", "\n", "This is an analytic, or closed-form, result. The equation directly returns the answer;\n", "no time-stepping simulation is required.\n", "\n", "## Translate the relationships without changing them\n", "\n", "The Python should preserve the assumptions and the direction of each relationship.\n", "It also needs to reject inputs that have no meaning in this model." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "from math import sqrt\n", "\n", "height_metres = 10.0\n", "gravity_metres_per_second_squared = 9.81\n", "assert height_metres >= 0\n", "assert gravity_metres_per_second_squared > 0\n", "landing_time_seconds = sqrt(2 * height_metres / gravity_metres_per_second_squared)\n", "print(f\"Landing time: {landing_time_seconds:.3f} s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The Python is short because the physical reasoning has already done the hard work." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "from math import sqrt\n", "\n", "\n", "def landing_time(\n", " height_metres: float,\n", " gravity_metres_per_second_squared: float = 9.81,\n", ") -> float:\n", " if height_metres < 0:\n", " raise ValueError(\"height must not be negative\")\n", "\n", " if gravity_metres_per_second_squared <= 0:\n", " raise ValueError(\"gravity magnitude must be positive\")\n", "\n", " return sqrt((2 * height_metres) / gravity_metres_per_second_squared)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "At a height of $h=10\\,\\mathrm{m}$\n", "and with $g=9.81\\,\\mathrm{m/s^2}$,\n", "the result is about $1.43\\,\\mathrm{s}$.\n", "At $h=40\\,\\mathrm{m}$,\n", "the result is about $2.86\\,\\mathrm{s}$.\n", "Multiplying the height by four multiplies the time by two because time depends on the\n", "square root of height. Code makes that scaling law easy to check across many values,\n", "but the equation explains why it happens.\n", "\n", "## What code adds\n", "\n", "A calculator can evaluate one case. A small program can answer a family of questions:\n", "\n", "- How does landing time change as height changes?\n", "- Which inputs dominate the uncertainty in the result?\n", "- Does the implementation reproduce known limiting cases?\n", "- How large is the difference if local gravity is used instead of\n", " $9.81\\,\\mathrm{m/s^2}$?\n", "- When does ignoring air resistance become visibly inadequate?\n", "\n", "Code also lets a claim become an executable check. Three useful checks are:" ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "def close_enough(actual: float, expected: float) -> bool:\n", " return abs(actual - expected) < 1e-12\n", "\n", "\n", "if not close_enough(landing_time(0), 0):\n", " raise AssertionError(\"zero height must give zero landing time\")\n", "\n", "short_fall = landing_time(10)\n", "four_times_higher = landing_time(40)\n", "\n", "if not close_enough(four_times_higher, 2 * short_fall):\n", " raise AssertionError(\"four times the height must double the time\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "These checks do not confirm that air resistance is negligible. They confirm that the\n", "code follows this particular equation. OpenStax distinguishes a model from reality in\n", "the same way: a model represents selected aspects of a system, and calculations or\n", "computer simulations still need comparison with observation and experiment.\n", "\n", "## Equations, algorithms, and simulations are different\n", "\n", "An equation states a relationship. An algorithm gives finite steps for evaluating or\n", "approximating a relationship. A simulation repeatedly advances a model's state. The\n", "free-fall problem above has an equation that can be solved directly, so wrapping it in a\n", "time loop would add numerical error without answering a new question.\n", "\n", "A loop becomes useful when the acceleration changes with position, velocity, or time.\n", "For example, an air-drag model may need current velocity to compute the next\n", "acceleration. Then the state contains position and velocity, and a numerical method\n", "approximates their evolution in small time intervals. That method introduces choices:\n", "step size, update order, stopping conditions, and an error check. More code is not\n", "automatically more physics; it often means more assumptions to document.\n", "\n", "## Verification is not validation\n", "\n", "Verification asks whether the implementation solves the stated mathematical problem\n", "correctly. The zero-height and four-times-height checks are verification evidence.\n", "Validation asks whether the stated mathematical problem adequately represents the\n", "physical fall. Comparing predicted landing times with measurements is validation\n", "evidence.\n", "\n", "A verified program can still be a poor description of a feather because the no-drag\n", "assumption is inappropriate. A program can also agree with one experiment for the\n", "wrong reason if one error happens to cancel another. Reliable work therefore records\n", "both kinds of evidence instead of treating a plausible graph as proof.\n", "\n", "## The boundary of the example\n", "\n", "The constant-acceleration result is appropriate only while several approximations are\n", "reasonable. The change in altitude must be small enough that gravity is nearly\n", "constant. Air forces must be negligible for the desired accuracy. The object must be\n", "released rather than thrown, unless an initial-velocity term is added. The ground must\n", "be represented by the chosen zero of height.\n", "\n", "Those qualifications are not apologies for the model. They are part of the model.\n", "Computational physics becomes powerful when assumptions, equations, implementation,\n", "and checks remain visible together. That visibility is the real reason code can make\n", "physics easier.\n", "\n", "## Repetition should reveal structure\n", "\n", "The most valuable repetition is not printing hundreds of nearly identical numbers. It\n", "is changing one controlled input and looking for a pattern predicted by the model. For\n", "the fall-time relationship, a useful program can evaluate a sequence of heights and\n", "compare every result with the square-root scaling law. The table should retain height\n", "and time units, and it should say that the values assume release from rest without drag.\n", "That context keeps the output from being mistaken for universal measured data.\n", "\n", "The same habit scales to harder problems. A parameter sweep can reveal a resonance, a\n", "conservation check can expose accumulated numerical error, and repeated calculations\n", "can propagate uncertain inputs into a distribution of outputs. In each case, decide on\n", "the physical question and diagnostic before producing data. Otherwise computation can\n", "generate a large result set without increasing understanding.\n", "\n", "Code is also a record of reasoning. Another reader can see the constants, reproduce the\n", "calculation, change an assumption, and inspect the checks. Reproducibility does not make\n", "the conclusion correct by itself, but it makes mistakes easier to locate and claims\n", "easier to challenge. That is a more important advantage than speed alone.\n", "\n", "## A worked Python notebook\n", "\n", "An object is released from rest above level ground. Compare the landing times from several heights. The question for this calculation is precise: how does **fall time**\n", "change when we change **release height**, 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", "Air resistance is omitted and the gravitational field is uniform. 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", "1 to 40 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 release height, expressed in m. The output is\n", "fall time, expressed in s. 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", "$$\n", "f(q) = 0.451523640985731 \\sqrt{q}\n", "$$\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 10 m, the expression gives 1.42784 s. 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(\"sqrt(2*q/9.81)\", locals={\"q\": q})\n", "reference_input = 10.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted fall time:\", reference_output, \"s\")\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", "0.451524 s. The midpoint, at 20.5 m, gives\n", "2.04436 s. The upper endpoint gives 2.85569 s.\n", "Read these together with the assumptions: they describe different choices of release height\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([1.0, 20.5, 40.0])\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} s\")\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 release height; the vertical axis shows fall time. 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(1.0, 40.0, 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(\"Release height (m)\")\n", "ax.set_ylabel(\"Fall time (s)\")\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", "fall time change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 10 m is\n", "0.0713922 in those ratio units. Locally, the output increases 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", "$$\n", "f'(q) = \\frac{0.225761820492865}{\\sqrt{q}}\n", "$$\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 = 10.195\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, \"s\")\n", "print(\"Full expression:\", full_prediction, \"s\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 10.195 m. The tangent estimate is\n", "1.44176 s, while the full expression gives\n", "1.4417 s. 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 = 39.0\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", "release height inside the stated interval, predict whether fall time 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: Air resistance is omitted and the gravitational field is uniform. 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 s:\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(40.0) - evaluate(1.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(1.0, 40.0, 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", "- [OpenStax, *University Physics Volume 1*, Section 1.1, \"The Scope and Scale of Physics\"](https://openstax.org/books/university-physics-volume-1/pages/1-1-the-scope-and-scale-of-physics).\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", "- [Benjamin Crowell, *Light and Matter*, Chapter 0, \"Introduction and Review\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 1, \"Introduction\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter1/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "why-physics-needs-code.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/why-physics-needs-code", "source_sha256": "d55e023c8abdc6a01b9e284826e457a4e1a635c5f417c105fea4d49f30950d76" } }, "nbformat": 4, "nbformat_minor": 5 }