{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Momentum conservation tests\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/momentum-conservation-tests). 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": [ "Momentum conservation turns a physical law into a precise consistency check. The check\n", "is not merely \"momentum before equals momentum after.\" It must include a declared system,\n", "one reference frame, a time interval, and every external impulse that crosses the system\n", "boundary.\n", "\n", "## The quantity to test\n", "\n", "For fixed-mass classical particles, total momentum is\n", "\n", "\n", "$$\n", "\\mathbf P=\\sum_{i=1}^{N}m_i\\mathbf v_i\n", "$$\n", "\n", "\n", "Across an interval from\n", "$t_a$ to\n", "$t_b$, the balance law is\n", "\n", "\n", "$$\n", "\\mathbf P(t_b)-\\mathbf P(t_a)=\\int_{t_a}^{t_b}\\mathbf F_{\\mathrm{ext}}(t)\\,dt=\\mathbf J_{\\mathrm{ext}}\n", "$$\n", "\n", "\n", "Define the residual\n", "\n", "\n", "$$\n", "\\mathbf r_P=\\mathbf P_b-\\mathbf P_a-\\mathbf J_{\\mathrm{ext}}\n", "$$\n", "\n", "\n", "The ideal model predicts a zero vector. In computed or measured data, the residual is\n", "compared with stated numerical and experimental uncertainty. Testing only its magnitude\n", "loses directional information, so record components as well as a norm." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "masses = np.array([2.0, 3.0])\n", "initial = np.array([[4.0, 0.0], [0.0, 0.0]])\n", "final = np.array([[1.6, 0.0], [1.6, 0.0]])\n", "external_impulse = np.zeros(2)\n", "residual = np.sum(masses[:, None] * (final - initial), axis=0) - external_impulse\n", "print(\"Momentum residual:\", residual)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Every term has momentum units. A force accidentally added without multiplying by time\n", "is a dimensional error that a plausible small decimal cannot repair.\n", "\n", "## Define the system before evaluating it\n", "\n", "A cart bouncing from a wall does not conserve the cart's momentum. The wall exerts an\n", "external impulse on a cart-only system. If the system includes cart, wall, Earth, and\n", "their supports, the opposite momentum appears elsewhere, although Earth's velocity\n", "change may be imperceptibly small because its mass is enormous.\n", "\n", "Likewise, two colliding carts may be nearly isolated horizontally while gravity and\n", "normal force act vertically. Momentum can be tested component by component. Cancellation\n", "of vertical external forces does not mean those forces are absent; it means their net\n", "impulse over the interval is zero within the model.\n", "\n", "Changing membership invalidates a simple before-and-after sum. Fuel leaving a rocket,\n", "rain entering a cart, or particles deleted at a boundary carry momentum. Use a control-\n", "volume balance that includes momentum flux, or retain the crossing bodies in the selected\n", "closed system.\n", "\n", "## A useful tolerance\n", "\n", "Exact equality is generally inappropriate for floating-point or measured inputs. A\n", "combined tolerance can be written as\n", "\n", "\n", "$$\n", "\\|\\mathbf r_P\\|\\le\\varepsilon_{\\mathrm{abs}}+\\varepsilon_{\\mathrm{rel}}P_{\\mathrm{scale}}\n", "$$\n", "\n", "\n", "The absolute part handles values near zero. The relative part scales with a characteristic\n", "momentum, which might be the largest of initial, final, impulse, or summed individual\n", "momentum magnitudes. Using only the net initial momentum as a scale fails when large\n", "opposing momenta cancel to nearly zero.\n", "\n", "Tolerances should follow from input precision, algorithm error, and the decision being\n", "made. Choosing a wide tolerance after seeing a failure hides evidence. Record the policy\n", "before evaluating fixtures, and show residual-to-scale ratio rather than only pass or\n", "fail." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "scale = max(\n", " np.sum(masses * np.linalg.norm(initial, axis=1)),\n", " np.sum(masses * np.linalg.norm(final, axis=1)),\n", " np.linalg.norm(external_impulse),\n", " 1.0,\n", ")\n", "allowed_residual = 1e-12 + 1e-10 * scale\n", "assert np.linalg.norm(residual) <= allowed_residual\n", "print(\"Allowed residual (kg m/s):\", allowed_residual)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "## Core fixtures\n", "\n", "Begin with cases whose answers are known independently.\n", "\n", "For two isolated equal masses in a one-dimensional elastic collision, exchanging incoming\n", "velocities preserves momentum. For a perfectly inelastic collision, the shared velocity\n", "\n", "\n", "$$\n", "v_f=\\frac{m_1v_{1i}+m_2v_{2i}}{m_1+m_2}\n", "$$\n", "\n", "\n", "also preserves momentum while kinetic energy usually decreases. Both fixtures are\n", "important because a test suite that checks only elastic cases may accidentally couple\n", "momentum to an energy assumption.\n", "\n", "An internal explosion gives a different fixture. Two pieces initially at rest can fly\n", "apart with momenta that sum to zero. Kinetic energy increases from an internal store,\n", "but total momentum remains constant. This demonstrates again that momentum and kinetic\n", "energy test different properties.\n", "\n", "Test a known external impulse too. A constant external force over duration\n", "$\\Delta t$ should give\n", "\n", "\n", "$$\n", "\\Delta\\mathbf P=\\mathbf F_{\\mathrm{ext}}\\Delta t\n", "$$\n", "\n", "\n", "This catches a checker that always expects raw momentum change to be zero.\n", "\n", "## Property-based variations\n", "\n", "After fixed examples, vary masses, velocities, coordinate directions, and storage order\n", "over valid ranges. Positive masses and finite vectors are prerequisites. The results\n", "should obey transformations that the physics predicts.\n", "\n", "If every input velocity receives the same Galilean shift\n", "$\\mathbf V$, initial and\n", "final total momenta each shift by total mass times that velocity. Their balance residual\n", "remains unchanged for a fixed-mass system. Rotating every vector and force should rotate\n", "the residual without changing its magnitude. Permuting body storage should not change\n", "the physical total beyond documented summation effects." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "for angle in np.linspace(0, 2 * np.pi, 9):\n", " c, s = np.cos(angle), np.sin(angle)\n", " rotation = np.array([[c, -s], [s, c]])\n", " rotated_residual = np.sum(\n", " masses[:, None] * ((final - initial) @ rotation.T), axis=0\n", " )\n", " assert np.allclose(rotated_residual, rotation @ residual)\n", "print(\"Rotated cases passed.\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Random cases do not replace analytic edge cases. Include zero relative velocity, one body\n", "much heavier than the other, equal and opposite momenta, a stationary center of mass,\n", "and inputs near the supported numerical scale.\n", "\n", "## Internal pair-force checks\n", "\n", "In a many-body update, total momentum conservation often depends on pair forces being\n", "equal and opposite:\n", "\n", "\n", "$$\n", "\\mathbf F_{ij}+\\mathbf F_{ji}=\\mathbf0\n", "$$\n", "\n", "\n", "Check this locally before integrating. Generate both force accumulations from the same\n", "pair evaluation. If one side is recomputed after another body has moved, array order can\n", "create momentum drift.\n", "\n", "Pair-force cancellation alone does not prove trajectories are correct. A code path could\n", "apply the wrong equal-and-opposite force and still conserve total momentum. Compare force\n", "magnitude and direction with analytic cases, and check energy or angular momentum when\n", "the model predicts them.\n", "\n", "## One-step and long-interval tests\n", "\n", "A one-step test localizes faults. Record state before one update, applied external\n", "impulse, state afterward, and the residual. A long-interval test reveals accumulation,\n", "biased rounding, intermittent contact errors, or boundary events.\n", "\n", "Report maximum residual, final residual, and trend. Oscillating roundoff around zero is\n", "different from monotonic drift. Do not sample only at specially aligned times where an\n", "error happens to cancel.\n", "\n", "Reduce the time step and compare at the same physical duration. For a method of known\n", "order, the residual should show an interpretable convergence pattern until floating-\n", "point or event-location error dominates. If it remains unchanged, the problem may be a\n", "missing impulse or inconsistent system definition rather than time integration.\n", "\n", "## Collision-event diagnostics\n", "\n", "At each collision, store the bodies involved, contact normal, pre-contact relative\n", "velocity, applied impulse, and post-contact state. An isolated pair impulse should add\n", "opposite momentum to the two bodies. Reject impulses applied to separating contacts\n", "unless the contact model explicitly justifies them.\n", "\n", "Penetration correction deserves a separate ledger. Moving positions without changing\n", "velocity does not directly alter calculated momentum, yet a later velocity correction\n", "can. If a position fix applies effective impulse, include it rather than allowing a hidden\n", "source.\n", "\n", "Simultaneous contacts can make sequential pair resolution order-dependent. Permute the\n", "contact order and compare; large changes indicate that a coupled solve or better stated\n", "approximation is needed.\n", "\n", "## Measurement-based checks\n", "\n", "Experimental momentum uses measured masses and velocities, each with uncertainty. The\n", "uncertainty in a one-dimensional product is approximately\n", "\n", "\n", "$$\n", "\\sigma_p^2\\approx v^2\\sigma_m^2+m^2\\sigma_v^2\n", "$$\n", "\n", "\n", "when mass and velocity errors are independent. Correlated errors require covariance\n", "terms. A residual consistent with zero within uncertainty supports conservation; it does\n", "not prove exact equality or validate every model assumption.\n", "\n", "Velocity estimated from position frames can be noisy, and camera calibration or frame\n", "timing can introduce shared bias. Compare results across analysis choices and avoid\n", "claiming physical momentum loss from a residual smaller than measurement capability.\n", "\n", "## Limits of the particle-only test\n", "\n", "Electromagnetic systems can store momentum in fields, so summing only mechanical\n", "particle momentum may omit a real contribution. Radiation can carry momentum out of the\n", "selected region. At relativistic speeds, use relativistic particle momentum rather than\n", "$m\\mathbf v$.\n", "\n", "These cases do not invalidate conservation. They show that the ledger must contain every\n", "relevant momentum store and flux for the stated physical regime.\n", "\n", "## Reliability checks\n", "\n", "- Declare system membership, frame, axes, and comparison interval.\n", "- Calculate totals from unrounded authoritative states.\n", "- Include signed external impulse and boundary momentum flux.\n", "- Preserve and inspect the vector residual component by component.\n", "- Scale tolerances with individual momenta, not only a nearly cancelling net value.\n", "- Test elastic, inelastic, explosive, and externally forced fixtures.\n", "- Verify pair-force cancellation and force correctness separately.\n", "- Check one-step behavior, long-time drift, and time-step convergence.\n", "- Keep momentum checks independent from kinetic-energy checks.\n", "- Include field or relativistic momentum when the regime requires it.\n", "\n", "A good conservation test is an accounting proof with an error budget. When it fails, its\n", "stored residual, boundary events, and impulse ledger should make the missing momentum\n", "traceable rather than merely producing a red indicator.\n", "\n", "## A worked Python notebook\n", "\n", "Boost a two-body system with total mass 5 kilograms and initial total momentum 8 kilogram metres per second. The question for this calculation is precise: how does **total momentum after boost**\n", "change when we change **common velocity boost**, 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", "The same boost applies to every body; a conserved total can have different numerical values in different frames. 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", "-3 to 3 m/s. 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 common velocity boost, expressed in m/s. The output is\n", "total momentum after boost, expressed in kg m/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", "$$\n", "f(q) = 5 q + 8\n", "$$\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 1 m/s, the expression gives 13 kg m/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(\"8+5*q\", locals={\"q\": q})\n", "reference_input = 1.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m/s\")\n", "print(\"Predicted total momentum after boost:\", reference_output, \"kg m/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", "-7 kg m/s. The midpoint, at 0 m/s, gives\n", "8 kg m/s. The upper endpoint gives 23 kg m/s.\n", "Read these together with the assumptions: they describe different choices of common velocity boost\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([-3.0, 0.0, 3.0])\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/s -> {output_value:.6g} kg m/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 common velocity boost; the vertical axis shows total momentum after boost. 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(-3.0, 3.0, 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(\"Common velocity boost (m/s)\")\n", "ax.set_ylabel(\"Total momentum after boost (kg m/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", "total momentum after boost change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 1 m/s is\n", "5 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", "$$\n", "f'(q) = 5\n", "$$\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 = 1.03\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, \"kg m/s\")\n", "print(\"Full expression:\", full_prediction, \"kg m/s\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"kg m/s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 1.03 m/s. The tangent estimate is\n", "13.15 kg m/s, while the full expression gives\n", "13.15 kg m/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 = 6.0\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", "common velocity boost inside the stated interval, predict whether total momentum after boost 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: The same boost applies to every body; a conserved total can have different numerical values in different frames. 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 kg m/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(3.0) - evaluate(-3.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(-3.0, 3.0, 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 9.2, \"Impulse and Collisions\"](https://openstax.org/books/university-physics-volume-1/pages/9-2-impulse-and-collisions).\n", "- [OpenStax, *University Physics Volume 1*, Section 9.3, \"Conservation of Linear Momentum\"](https://openstax.org/books/university-physics-volume-1/pages/9-3-conservation-of-linear-momentum).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 10, Sections 10.7–10.8](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter10/).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 15, \"Collision Theory\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter15/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 14, \"Conservation of Momentum\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Conservation of Momentum\"](https://physics.info/momentum-conservation/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "momentum-conservation-tests.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/momentum-conservation-tests", "source_sha256": "01149a24c8abcf385447e0832e73996331907ce22bcbbed026ff7d43b6038284" } }, "nbformat": 4, "nbformat_minor": 5 }