{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Kinetic energy from data\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/kinetic-energy-from-data). 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": [ "Translational kinetic energy turns mass and velocity into a scalar measure of energy of\n", "motion. Direction disappears from the final value, but the original velocity vector,\n", "reference frame, time, units, and measurement method remain essential. A speed copied\n", "from a rounded display is weaker evidence than the underlying timestamped state.\n", "\n", "## Classical translational kinetic energy\n", "\n", "For a particle or body whose center of mass moves at speed\n", "$v$ in a stated inertial frame,\n", "\n", "\n", "$$\n", "K=\\frac12mv^2=\\frac12m\\mathbf v\\cdot\\mathbf v\n", "$$\n", "\n", "\n", "Mass must be nonnegative in this ordinary model, and velocity components must share one\n", "basis and unit. Because a vector dotted with itself is nonnegative, classical kinetic\n", "energy is nonnegative. It is zero when velocity is zero in the selected frame.\n", "\n", "In SI base units,\n", "\n", "\n", "$$\n", "[K]=\\mathrm{kg\\,m^2/s^2}=\\mathrm J\n", "$$\n", "\n", "\n", "Reversing velocity leaves energy unchanged:\n", "$K(-\\mathbf v)=K(\\mathbf v)$.\n", "Momentum reverses, so kinetic energy alone cannot reconstruct the state." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "\n", "def kinetic_energy(mass, velocity):\n", " velocity = np.asarray(velocity, dtype=float)\n", " if mass < 0 or not np.all(np.isfinite(velocity)):\n", " raise ValueError(\"Use nonnegative mass and finite velocity.\")\n", " return 0.5 * mass * np.dot(velocity, velocity)\n", "\n", "\n", "print(f\"Energy: {kinetic_energy(2.0, [3.0, 4.0]):.1f} J\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "For ordinary measured matter, negative mass input is invalid. A zero mass with finite\n", "classical velocity produces zero in the algebra but does not model a photon; relativistic\n", "massless particles require a different energy relation.\n", "\n", "## Scaling predictions\n", "\n", "At fixed speed, doubling mass doubles kinetic energy. At fixed mass, doubling speed\n", "quadruples it:\n", "\n", "\n", "$$\n", "K(cm,v)=cK(m,v),\\qquad K(m,cv)=c^2K(m,v)\n", "$$\n", "\n", "\n", "These relations are useful checks because they do not depend on a particular numeric\n", "example. They also explain why small speed errors can become more important after\n", "squaring.\n", "\n", "For mass\n", "$m=2.0\\,\\mathrm{kg}$ and velocity\n", "$\\mathbf v=(3,4)\\,\\mathrm{m/s}$,\n", "speed is $5\\,\\mathrm{m/s}$\n", "and\n", "\n", "\n", "$$\n", "K=\\frac12(2.0)(5^2)=25\\,\\mathrm J\n", "$$\n", "\n", "\n", "## Kinetic energy depends on reference frame\n", "\n", "If another inertial frame moves at constant velocity\n", "$\\mathbf V$ relative to the first,\n", "classical velocity transforms as\n", "$\\mathbf v'=\\mathbf v-\\mathbf V$.\n", "Therefore\n", "\n", "\n", "$$\n", "K'=\\frac12m|\\mathbf v-\\mathbf V|^2\n", "$$\n", "\n", "\n", "Expanding,\n", "\n", "\n", "$$\n", "K'=K-m\\mathbf V\\cdot\\mathbf v+\\frac12mV^2\n", "$$\n", "\n", "\n", "Kinetic energy is not Galilean invariant. A passenger at rest in a train has zero\n", "kinetic energy relative to the train but nonzero kinetic energy relative to the ground.\n", "This is not a contradiction: energy accounting, work, and system boundaries must use a\n", "consistent frame." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "measured_velocity = np.array([3.0, 4.0])\n", "frame_velocity = np.array([3.0, 0.0])\n", "relative_velocity = measured_velocity - frame_velocity\n", "print(f\"Energy in moving frame: {kinetic_energy(2.0, relative_velocity):.1f} J\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "The frame velocity must be expressed in the same basis, units, and timestamp convention\n", "as the measured velocity.\n", "\n", "## A system of particles\n", "\n", "For particles labeled by\n", "$i$,\n", "\n", "\n", "$$\n", "K_{\\mathrm{total}}=\\sum_i\\frac12m_i|\\mathbf v_i|^2\n", "$$\n", "\n", "\n", "Let total mass be\n", "$M=\\sum_i m_i$ and center-of-mass velocity\n", "\n", "\n", "$$\n", "\\mathbf V_{\\mathrm{CM}}=\\frac{1}{M}\\sum_i m_i\\mathbf v_i\n", "$$\n", "\n", "\n", "Writing each velocity as center-of-mass velocity plus a relative part gives\n", "\n", "\n", "$$\n", "K_{\\mathrm{total}}=\\frac12M V_{\\mathrm{CM}}^2+\\sum_i\\frac12m_i|\\mathbf v_i-\\mathbf V_{\\mathrm{CM}}|^2\n", "$$\n", "\n", "\n", "The first term is bulk translational energy. The second is kinetic energy relative to\n", "the center of mass. For a gas, that relative microscopic motion contributes to internal\n", "energy, but internal energy also depends on molecular rotation, vibration, and\n", "interactions. It should not be identified with one unspecific kinetic-energy sum." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "masses = np.array([1.0, 2.0])\n", "velocities = np.array([[3.0, 0.0], [-1.0, 0.0]])\n", "com_velocity = np.sum(masses[:, None] * velocities, axis=0) / masses.sum()\n", "bulk_energy = kinetic_energy(masses.sum(), com_velocity)\n", "relative_energy = sum(\n", " kinetic_energy(m, v - com_velocity) for m, v in zip(masses, velocities)\n", ")\n", "total = sum(kinetic_energy(m, v) for m, v in zip(masses, velocities))\n", "assert np.isclose(total, bulk_energy + relative_energy)\n", "print(\"Total, bulk, relative energies (J):\", total, bulk_energy, relative_energy)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "The two terms should add back to the direct particle sum within floating-point\n", "tolerance.\n", "\n", "## Rotation is another energy store\n", "\n", "For a rigid body, center-of-mass translation is not the whole kinetic energy. Rotation\n", "about the center of mass contributes\n", "\n", "\n", "$$\n", "K_{\\mathrm{rot}}=\\frac12I\\omega^2\n", "$$\n", "\n", "\n", "for a simple fixed-axis case. The full rigid-body relation uses the inertia tensor and\n", "angular-velocity vector. A rolling object can have both translational and rotational\n", "kinetic energy. Applying only\n", "$\\tfrac12Mv_{\\mathrm{CM}}^2$\n", "would omit the rotation.\n", "\n", "Deformation and vibration can add further kinetic terms. The representation must match\n", "the system's modeled degrees of freedom.\n", "\n", "## Deriving velocity from position samples\n", "\n", "Position data does not directly contain instantaneous velocity. For equally spaced\n", "samples, an interior centered estimate is\n", "\n", "\n", "$$\n", "\\mathbf v(t_i)\\approx\\frac{\\mathbf r(t_{i+1})-\\mathbf r(t_{i-1})}{t_{i+1}-t_{i-1}}\n", "$$\n", "\n", "\n", "Use actual timestamps when spacing is uneven. A forward difference assigns an interval\n", "average to an endpoint unless a more careful time convention is stated. Differentiation\n", "amplifies high-frequency measurement noise, and squaring speed introduces positive bias:\n", "zero-mean velocity noise does not generally average to zero energy error." ] }, { "cell_type": "code", "id": "code-009", "metadata": {}, "source": [ "times = np.linspace(0, 2, 21)\n", "positions = 3 * times + 0.5 * times**2\n", "velocities = np.gradient(positions, times, edge_order=2)\n", "energies = 0.5 * 2.0 * velocities**2\n", "print(\"Energy at 1 s (J):\", energies[10])\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "Filtering or curve fitting can reduce noise but adds assumptions and may erase rapid\n", "motion. Report the derivative or fit method, window, boundary handling, and sampling\n", "rate.\n", "\n", "## Uncertainty propagation\n", "\n", "For independent small uncertainties in scalar mass and speed,\n", "\n", "\n", "$$\n", "\\sigma_K^2\\approx\\left(\\frac12v^2\\sigma_m\\right)^2+(mv\\sigma_v)^2\n", "$$\n", "\n", "\n", "If mass and speed estimates are correlated, add the covariance term. For vector velocity\n", "with covariance matrix\n", "$\\boldsymbol\\Sigma_v$,\n", "linear propagation away from zero velocity uses gradient\n", "$\\partial K/\\partial\\mathbf v=m\\mathbf v$.\n", "\n", "Near zero velocity, first-order propagation can be misleading because that gradient\n", "vanishes while squared noise still produces positive estimated energy. Monte Carlo\n", "propagation or a measurement-specific statistical model can be more appropriate.\n", "\n", "## Work-energy check\n", "\n", "For a constant-mass particle,\n", "\n", "\n", "$$\n", "\\Delta K=W_{\\mathrm{net}}=\\int\\mathbf F_{\\mathrm{net}}\\cdot d\\mathbf r\n", "$$\n", "\n", "\n", "Compute the kinetic-energy difference from endpoint velocity data and net work from\n", "force-position data independently. Agreement checks timestamps, force signs, units, and\n", "path integration. If the data describe an extended or changing-mass system, clarify\n", "which work-energy statement applies before treating disagreement as error.\n", "\n", "## Relativistic limit\n", "\n", "Classical kinetic energy is accurate when speed is much smaller than the speed of light.\n", "Special relativity uses\n", "\n", "\n", "$$\n", "K=(\\gamma-1)mc^2,\\qquad \\gamma=\\frac{1}{\\sqrt{1-v^2/c^2}}\n", "$$\n", "\n", "\n", "For $v\\ll c$, its\n", "series begins with\n", "$\\tfrac12mv^2$. The classical\n", "formula increasingly underestimates energy as speed approaches\n", "$c$ and cannot describe a massive body\n", "at or above that speed.\n", "\n", "## Validation checks\n", "\n", "- Zero velocity in the selected frame gives zero classical kinetic energy.\n", "- Reversing velocity leaves energy unchanged while reversing momentum.\n", "- Doubling speed quadruples energy; doubling mass doubles it.\n", "- A three-four component velocity gives the expected five-unit speed.\n", "- Frame transformation changes energy according to the transformed velocity.\n", "- Particle energy equals bulk center-of-mass plus relative kinetic energy.\n", "- Translational and rotational stores are not double-counted or omitted.\n", "- Velocity estimates use actual timestamps and report their differentiation method.\n", "- Uncertainty accounts for squaring, covariance, and near-zero bias where relevant.\n", "- Net work agrees with kinetic-energy change under the theorem's assumptions.\n", "\n", "Kinetic energy is easy to calculate only after the data's meaning is secure. Mass,\n", "velocity, frame, timestamp, modeled degrees of freedom, and uncertainty determine what\n", "the scalar result actually represents.\n", "\n", "## A worked Python notebook\n", "\n", "Compare kinetic energies for a 2 kilogram body moving at different speeds. The question for this calculation is precise: how does **kinetic energy**\n", "change when we change **speed**, 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", "Classical kinetic energy applies when speeds are small compared with light speed. 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 8 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 speed, expressed in m/s. The output is\n", "kinetic energy, expressed in J. 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", "$$\n", "f(q) = q^{2}\n", "$$\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 5 m/s, the expression gives 25 J. 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-011", "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(\"q**2\", locals={\"q\": q})\n", "reference_input = 5.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m/s\")\n", "print(\"Predicted kinetic energy:\", reference_output, \"J\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-012", "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 J. The midpoint, at 4 m/s, gives\n", "16 J. The upper endpoint gives 64 J.\n", "Read these together with the assumptions: they describe different choices of speed\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-013", "metadata": {}, "source": [ "evaluate = sp.lambdify(q, expression, modules=\"numpy\")\n", "inputs = np.array([0.0, 4.0, 8.0])\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} J\")\n", "assert np.all(np.isfinite(outputs))\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-014", "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 speed; the vertical axis shows kinetic energy. 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-015", "metadata": {}, "source": [ "input_grid = np.linspace(0.0, 8.0, 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(\"Speed (m/s)\")\n", "ax.set_ylabel(\"Kinetic energy (J)\")\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-016", "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", "kinetic energy change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 5 m/s is\n", "10 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", "$$\n", "f'(q) = 2 q\n", "$$\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-017", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 5.04\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, \"J\")\n", "print(\"Full expression:\", full_prediction, \"J\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"J\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-018", "metadata": {}, "source": [ "Here the changed input is 5.04 m/s. The tangent estimate is\n", "25.4 J, while the full expression gives\n", "25.4016 J. 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-019", "metadata": {}, "source": [ "input_span = 8.0\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-020", "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", "speed inside the stated interval, predict whether kinetic energy 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: Classical kinetic energy applies when speeds are small compared with light speed. 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 J:\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-021", "metadata": {}, "source": [ "slope_function = sp.lambdify(q, slope_expression, modules=\"numpy\")\n", "endpoint_change = float(evaluate(8.0) - evaluate(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 8.0, 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-022", "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 7.2, \"Kinetic Energy and the Work-Energy Theorem\"](https://openstax.org/books/university-physics-volume-1/pages/7-2-kinetic-energy-and-the-work-energy-theorem).\n", "- [OpenStax, *University Physics Volume 1*, Section 9.6, \"Center of Mass\"](https://openstax.org/books/university-physics-volume-1/pages/9-6-center-of-mass).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 13, Sections 13.1–13.3](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter13/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 14, \"Conservation of Energy\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Kinetic Energy\"](https://physics.info/energy-kinetic/).\n", "- [Tipler and Llewellyn, *Modern Physics*, Chapter 2, \"Special Relativity\"](https://web.pdx.edu/~pmoeck/books/Tipler_Llewellyn.pdf)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "kinetic-energy-from-data.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/kinetic-energy-from-data", "source_sha256": "29e0a979b57518b1a60e3f71a69b939ef289504aa020e2ed65b63edcfb1fb009" } }, "nbformat": 4, "nbformat_minor": 5 }