{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Linear momentum as state\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/linear-momentum-as-state). 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": [ "Linear momentum combines an object's inertia and motion into one vector. For a particle\n", "of constant mass moving much slower than light,\n", "\n", "\n", "$$\n", "\\mathbf p=m\\mathbf v\n", "$$\n", "\n", "\n", "The mass $m$ is a nonnegative scalar, velocity\n", "$\\mathbf v$ is a vector, and momentum\n", "$\\mathbf p$ points in the same\n", "direction as velocity when mass is positive. Its SI unit is kilogram metre per second,\n", "equivalently newton second.\n", "\n", "## Why momentum belongs in the state\n", "\n", "Position alone does not determine how motion continues. Two equal objects at the same\n", "place can have different futures if their velocities differ. Momentum records that\n", "motion while also including mass, which makes it especially convenient when objects of\n", "different masses interact.\n", "\n", "For constant mass, velocity and momentum contain equivalent information because\n", "\n", "\n", "$$\n", "\\mathbf v=\\frac{\\mathbf p}{m}\n", "$$\n", "\n", "\n", "Momentum is often the better system variable because total momentum is a direct vector\n", "sum. It is also the natural quantity changed by force and impulse. A state representation\n", "should not store velocity and momentum as independent authoritative fields unless it\n", "also enforces their consistency; otherwise they can silently disagree." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "state = {\"mass_kg\": 2.0, \"position_m\": 1.0, \"momentum_kg_m_per_s\": 6.0}\n", "velocity = state[\"momentum_kg_m_per_s\"] / state[\"mass_kg\"]\n", "print(f\"Velocity: {velocity:.1f} m/s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The division requires positive nonzero mass in this classical particle model. Massless\n", "particles are outside the formula's domain, and relativistic particles use a different\n", "momentum-velocity relationship.\n", "\n", "## Momentum is a vector\n", "\n", "In one dimension, a sign can represent direction. Choosing rightward as positive makes a\n", "left-moving object's momentum negative. In three dimensions, calculate components:\n", "\n", "\n", "$$\n", "\\mathbf p=(mv_x,mv_y,mv_z)\n", "$$\n", "\n", "\n", "Opposite momenta can cancel even though neither object is at rest. For two equal masses\n", "moving with equal speeds in opposite directions, total momentum is zero while total\n", "kinetic energy is positive. Momentum conservation therefore does not mean that every\n", "object keeps the same momentum, nor does zero total momentum mean no motion.\n", "\n", "Never add vector magnitudes when the directions differ. Add components first, then take\n", "the magnitude only if the question asks for it:\n", "\n", "\n", "$$\n", "\\mathbf P=\\sum_{i=1}^{N}\\mathbf p_i,\\qquad P=|\\mathbf P|\n", "$$\n", "\n", "\n", "Here $\\mathbf P$ is the system\n", "total and $P$ is a\n", "nonnegative scalar. These are not interchangeable.\n", "\n", "## Force is the rate of momentum change\n", "\n", "Newton's second law in its general momentum form is\n", "\n", "\n", "$$\n", "\\mathbf F_{\\mathrm{net}}=\\frac{d\\mathbf p}{dt}\n", "$$\n", "\n", "\n", "For a single constant-mass particle, substituting\n", "$\\mathbf p=m\\mathbf v$\n", "gives the familiar\n", "\n", "\n", "$$\n", "\\mathbf F_{\\mathrm{net}}=m\\mathbf a\n", "$$\n", "\n", "\n", "The momentum form should not be expanded naively as\n", "$m\\mathbf a+\\mathbf v\\,dm/dt$\n", "for a rocket without accounting for the momentum carried across the changing system\n", "boundary. Variable-mass problems require an explicit control volume or momentum-flux\n", "analysis.\n", "\n", "Integrating force over a time interval produces impulse:\n", "\n", "\n", "$$\n", "\\mathbf J=\\int_{t_1}^{t_2}\\mathbf F_{\\mathrm{net}}\\,dt=\\Delta\\mathbf p\n", "$$\n", "\n", "\n", "This relationship is valuable in collisions because the contact force may be large and\n", "brief while the total impulse remains measurable from the before-and-after momenta.\n", "\n", "## System momentum and internal exchange\n", "\n", "For a system of particles,\n", "\n", "\n", "$$\n", "\\mathbf P=\\sum_i m_i\\mathbf v_i\n", "$$\n", "\n", "\n", "Internal forces redistribute momentum between parts. Under the usual classical\n", "assumptions that internal interaction forces cancel in pairs, their contributions vanish\n", "from the system total, leaving\n", "\n", "\n", "$$\n", "\\frac{d\\mathbf P}{dt}=\\mathbf F_{\\mathrm{external}}\n", "$$\n", "\n", "\n", "If the net external impulse is zero over an interval, total momentum at the end equals\n", "total momentum at the beginning. The system does not have to be completely free of\n", "external forces; several external impulses can cancel. Conversely, a short interaction\n", "with a wall, track, person, expelled gas, or Earth can transfer momentum even if that\n", "external body barely moves visibly." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "masses = np.array([2.0, 3.0])\n", "velocities = np.array([[3.0, 0.0], [-1.0, 2.0]])\n", "total_momentum = np.sum(masses[:, None] * velocities, axis=0)\n", "print(\"Total momentum (kg m/s):\", total_momentum)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "The list of bodies defines the boundary. Record it with the result so another reader can\n", "tell whether a force is internal or external.\n", "\n", "## Reference-frame dependence\n", "\n", "Classical momentum depends on the inertial frame. If a frame moves at constant velocity\n", "$\\mathbf V$ relative to the original,\n", "then Galilean transformation gives\n", "\n", "\n", "$$\n", "\\mathbf v_i'=\\mathbf v_i-\\mathbf V\n", "$$\n", "\n", "\n", "and for a fixed-mass system,\n", "\n", "\n", "$$\n", "\\mathbf P'=\\mathbf P-M\\mathbf V\n", "$$\n", "\n", "\n", "where $M=\\sum_i m_i$. Numerical\n", "momentum values change between frames, but conservation holds in every inertial frame\n", "when the assumptions hold. Before comparing two results, confirm that their axes, origin\n", "motion, and time interval match.\n", "\n", "The center-of-momentum frame is the inertial frame in which total momentum is zero. For\n", "nonrelativistic fixed masses, it moves with the center-of-mass velocity\n", "\n", "\n", "$$\n", "\\mathbf V_{\\mathrm{cm}}=\\frac{\\mathbf P}{M}\n", "$$\n", "\n", "\n", "Individual bodies can still move rapidly in this frame.\n", "\n", "## Worked two-body calculation\n", "\n", "Take a\n", "$2\\,\\mathrm{kg}$ cart moving right at\n", "$3\\,\\mathrm{m/s}$ and a\n", "$1\\,\\mathrm{kg}$ cart moving left at\n", "$4\\,\\mathrm{m/s}$. With right\n", "positive,\n", "\n", "\n", "$$\n", "P=(2)(3)+(1)(-4)=2\\ \\mathrm{kg\\,m/s}\n", "$$\n", "\n", "\n", "The total points right even though one cart moves left. If the carts interact on an\n", "effectively isolated track, any acceptable outgoing velocities must produce the same\n", "total. Momentum alone may not determine both outgoing velocities; collision type or a\n", "coefficient of restitution supplies additional information.\n", "\n", "## Numerical updates and residuals\n", "\n", "For a computed system, update each body's momentum from forces and then evaluate the\n", "total. A simple step has the structure\n", "\n", "\n", "$$\n", "\\mathbf p_i^{n+1}=\\mathbf p_i^n+\\mathbf F_i^n\\Delta t\n", "$$\n", "\n", "\n", "This is first-order time integration, not an exact formula for changing force. Internal\n", "force pairs should be generated consistently so equal-and-opposite impulses cancel to\n", "roundoff. Computing the two sides through unrelated approximations can create artificial\n", "system momentum." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "momenta = masses[:, None] * velocities\n", "before = momenta.sum(axis=0)\n", "forces = np.array([[4.0, 0.0], [-4.0, 0.0]])\n", "dt = 0.1\n", "momenta += forces * dt\n", "external_impulse = np.zeros(2)\n", "residual = momenta.sum(axis=0) - before - external_impulse\n", "assert np.allclose(residual, 0.0)\n", "print(\"Momentum residual:\", residual)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "Compare the residual with a physically scaled absolute-plus-relative tolerance. Reduce\n", "the time step and verify convergence. A conserved total can still coexist with incorrect\n", "individual trajectories, so also test analytic cases, energy when appropriate, and\n", "contact conditions.\n", "\n", "## Domain and model limits\n", "\n", "The expression\n", "$\\mathbf p=m\\mathbf v$ is a\n", "nonrelativistic approximation. At speeds approaching the speed of light, momentum is\n", "\n", "\n", "$$\n", "\\mathbf p=\\gamma m\\mathbf v,\\qquad \\gamma=\\frac{1}{\\sqrt{1-v^2/c^2}}\n", "$$\n", "\n", "\n", "Momentum conservation remains fundamental, but the relationship to velocity changes.\n", "At ordinary mechanical speeds, the classical expression is accurate enough.\n", "\n", "## Reliability checks\n", "\n", "- Declare the inertial reference frame and coordinate directions.\n", "- Preserve momentum as a vector with units.\n", "- Add vector components, not magnitudes.\n", "- Define the system boundary before labeling forces internal or external.\n", "- Compare momentum change with external impulse, not automatically with zero.\n", "- Keep velocity and momentum state fields consistent.\n", "- Treat changing-mass systems with momentum flux across the boundary.\n", "- Reduce time step and inspect component-wise conservation residuals.\n", "- Use relativistic momentum when the classical speed approximation fails.\n", "\n", "Momentum is useful state because it connects microscopic exchanges to a simple system\n", "balance. It can change dramatically for one object while remaining exactly accounted\n", "for across a well-defined isolated system.\n", "\n", "## A worked Python notebook\n", "\n", "Store the momentum of a 2 kilogram body and inspect its relation to velocity. The question for this calculation is precise: how does **momentum**\n", "change when we change **velocity**, 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 mass is fixed; changing mass requires an explicit open-system momentum balance. 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", "-5 to 5 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 velocity, expressed in m/s. The output is\n", "momentum, 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", "$$\n", "f(q) = 2 q\n", "$$\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 3 m/s, the expression gives 6 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(\"2*q\", locals={\"q\": q})\n", "reference_input = 3.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m/s\")\n", "print(\"Predicted momentum:\", 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", "-10 kg m/s. The midpoint, at 0 m/s, gives\n", "0 kg m/s. The upper endpoint gives 10 kg m/s.\n", "Read these together with the assumptions: they describe different choices of velocity\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([-5.0, 0.0, 5.0])\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 velocity; the vertical axis shows momentum. 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(-5.0, 5.0, 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(\"Velocity (m/s)\")\n", "ax.set_ylabel(\"Momentum (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", "momentum change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 3 m/s is\n", "2 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", "$$\n", "f'(q) = 2\n", "$$\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 = 3.05\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 3.05 m/s. The tangent estimate is\n", "6.1 kg m/s, while the full expression gives\n", "6.1 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 = 10.0\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", "velocity inside the stated interval, predict whether momentum 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 mass is fixed; changing mass requires an explicit open-system momentum balance. 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(5.0) - evaluate(-5.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(-5.0, 5.0, 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 9.1, \"Linear Momentum\"](https://openstax.org/books/university-physics-volume-1/pages/9-1-linear-momentum).\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", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 10, \"Momentum, System of Particles, and Conservation of Momentum\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter10/).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 11, \"Reference Frames\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter11/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 14, \"Conservation of Momentum\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Momentum\"](https://physics.info/momentum/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "linear-momentum-as-state.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/linear-momentum-as-state", "source_sha256": "52c4e79ccf64b901368597daf777634dff860abaf010088004b46bca4bc79259" } }, "nbformat": 4, "nbformat_minor": 5 }