{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Escape velocity and energy\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/escape-velocity-energy). 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": [ "Escape speed is the minimum initial speed that lets an unpowered body reach arbitrarily\n", "large distance with no speed left over, under an ideal isolated two-body gravitational\n", "model. It is defined at a particular starting radius. It is not a universal speed and is\n", "not the same as the speed a real rocket must show at launch.\n", "\n", "## Gravitational potential energy\n", "\n", "For two point or spherical masses separated by center-to-center distance\n", "$r$, choose zero\n", "potential energy at infinite separation. Then\n", "\n", "\n", "$$\n", "U(r)=-\\frac{GMm}{r}\n", "$$\n", "\n", "\n", "The negative sign says a bound pair has less energy than the separated reference state.\n", "Work must be supplied to move the bodies infinitely far apart. Changing the arbitrary\n", "zero of potential would change numerical energy values, but not energy differences; the\n", "standard zero at infinity makes escape classification especially simple.\n", "\n", "For an orbiting test body of mass\n", "$m$ around a much larger source, total\n", "mechanical energy is\n", "\n", "\n", "$$\n", "E=\\frac12mv^2-\\frac{GMm}{r}\n", "$$\n", "\n", "\n", "In the ideal model, gravity is conservative, so this energy remains constant.\n", "\n", "## Deriving escape speed\n", "\n", "At the minimum escape threshold, the body approaches infinite radius with speed\n", "approaching zero. Its final kinetic and potential energy both approach zero, so its total\n", "energy is zero. Setting the initial energy to zero gives\n", "\n", "\n", "$$\n", "\\frac12mv_{\\mathrm{esc}}^2-\\frac{GMm}{r}=0\n", "$$\n", "\n", "\n", "Solving produces\n", "\n", "\n", "$$\n", "v_{\\mathrm{esc}}=\\sqrt{\\frac{2GM}{r}}=\\sqrt{\\frac{2\\mu}{r}}\n", "$$\n", "\n", "\n", "where $\\mu=GM$. The\n", "escaping body's mass cancels. A heavier and lighter projectile at the same position have\n", "the same ideal escape-speed threshold, although atmosphere and propulsion can affect\n", "real vehicles differently." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "mu, radius = 3.986004418e14, 6.371e6\n", "escape_speed = np.sqrt(2 * mu / radius)\n", "print(f\"Escape speed: {escape_speed/1000:.3f} km/s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "Dimensional analysis confirms the result: gravitational parameter has units of length\n", "cubed per time squared, division by radius gives speed squared, and the square root gives\n", "speed.\n", "\n", "## Specific energy makes classification direct\n", "\n", "Divide mechanical energy by the orbiting mass:\n", "\n", "\n", "$$\n", "\\varepsilon=\\frac{E}{m}=\\frac{v^2}{2}-\\frac{\\mu}{r}\n", "$$\n", "\n", "\n", "This quantity has units of energy per mass, equivalent to speed squared. Its sign\n", "classifies ideal Kepler motion:\n", "\n", "- $\\varepsilon<0$ means a\n", " bound ellipse, including a circle.\n", "- $\\varepsilon=0$ is the\n", " parabolic escape boundary.\n", "- $\\varepsilon>0$ means an\n", " unbound hyperbola with nonzero speed remaining at infinity.\n", "\n", "Exact equality is fragile with measured data and floating-point arithmetic. Classify a\n", "small interval around zero as near-threshold or indeterminate and report the tolerance." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "for speed in [8000.0, escape_speed, 12000.0]:\n", " energy = 0.5 * speed**2 - mu / radius\n", " kind = \"unbound\" if energy > 1e-6 else \"bound\" if energy < -1e-6 else \"boundary\"\n", " print(speed, energy, kind)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "The tolerance must carry the same units as specific energy and should reflect input\n", "uncertainty as well as numerical error.\n", "\n", "## Relation to circular speed\n", "\n", "Circular speed at the same radius is\n", "\n", "\n", "$$\n", "v_{\\mathrm c}=\\sqrt{\\frac{\\mu}{r}}\n", "$$\n", "\n", "\n", "Therefore\n", "\n", "\n", "$$\n", "v_{\\mathrm{esc}}=\\sqrt{2}\\,v_{\\mathrm c}\n", "$$\n", "\n", "\n", "The two speeds answer different questions. Circular speed requires a tangential\n", "direction and produces constant radius. Escape speed is an energy threshold based on\n", "the total speed. A speed between the two can describe a bound ellipse when directed\n", "tangentially.\n", "\n", "At circular speed, specific energy is\n", "\n", "\n", "$$\n", "\\varepsilon_{\\mathrm c}=-\\frac{\\mu}{2r}\n", "$$\n", "\n", "\n", "which is negative, as expected for a bound orbit.\n", "\n", "## Speed sets energy, direction sets the path\n", "\n", "At one position, changing velocity direction without changing magnitude leaves specific\n", "energy unchanged. It does change angular momentum\n", "\n", "\n", "$$\n", "\\mathbf h=\\mathbf r\\times\\mathbf v\n", "$$\n", "\n", "\n", "and therefore changes the orbit's shape and closest approach. An outward radial launch\n", "at escape speed follows a radial limiting path. A tangential launch at the same speed\n", "follows a parabola with nonzero angular momentum.\n", "\n", "Energy greater than or equal to zero does not guarantee that a body avoids the central\n", "surface. A velocity directed inward may collide before reaching large radius. Thus a\n", "complete classifier should report both energy class and collision status under the\n", "chosen body-radius model.\n", "\n", "## Remaining speed above the threshold\n", "\n", "For a positive-energy trajectory, far from the source the potential term approaches\n", "zero. The asymptotic speed satisfies\n", "\n", "\n", "$$\n", "v_\\infty=\\sqrt{v_0^2-v_{\\mathrm{esc}}^2}\n", "$$\n", "\n", "\n", "where both initial speed\n", "$v_0$ and escape speed are\n", "evaluated at the same starting radius in the ideal model. This is sometimes called\n", "hyperbolic excess speed." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "initial_speed = 12000.0\n", "excess_squared = initial_speed**2 - escape_speed**2\n", "if excess_squared < 0:\n", " raise ValueError(\"The object cannot escape without further energy input.\")\n", "print(f\"Speed at infinity: {np.sqrt(excess_squared):.3f} m/s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "## Launching from a planet\n", "\n", "At the surface of a spherical planet, use the distance from its center, not altitude\n", "alone. With planet radius\n", "$R$, the ideal surface value is\n", "\n", "\n", "$$\n", "v_{\\mathrm{esc}}=\\sqrt{\\frac{2GM}{R}}\n", "$$\n", "\n", "\n", "Because surface gravity is\n", "$g=GM/R^2$, the\n", "same result may be written\n", "\n", "\n", "$$\n", "v_{\\mathrm{esc}}=\\sqrt{2gR}\n", "$$\n", "\n", "\n", "This second form assumes the supplied\n", "$g$ corresponds\n", "to the same spherical, nonrotating model. Treating gravity as constant all the way to\n", "infinity would instead give an infinite required energy and is internally inconsistent.\n", "The inverse-square decrease must be used.\n", "\n", "Planet rotation can contribute initial inertial speed depending on latitude and launch\n", "direction. Atmosphere adds drag, and a real rocket adds energy gradually while losing\n", "mass. Consequently the ideal instantaneous surface escape speed is a useful energy\n", "benchmark, not a required ground-relative speed at one moment during a powered launch.\n", "\n", "## More than one gravitating body\n", "\n", "In a multi-body system, there is generally no single global escape-speed formula based\n", "on one radius. Potential is the sum of all source contributions and changes in time when\n", "sources move. A body may escape one planet while remaining bound to its star, or use a\n", "gravitational encounter to exchange energy in a chosen frame.\n", "\n", "Energy labels must name the system boundary and reference frame. Mechanical energy of a\n", "spacecraft relative to one moving planet is not generally conserved during a multi-body\n", "encounter, even though total energy of the full isolated system is conserved.\n", "\n", "## Checking a numerical escape experiment\n", "\n", "A finite-time trajectory cannot literally reach infinity. Use analytic energy sign as\n", "the ideal classification, then propagate only to check motion, collision, and numerical\n", "quality. Track specific energy and angular momentum at fixed physical times. Reduce the\n", "time step and compare the same outputs.\n", "\n", "Near the zero-energy boundary, tiny input or solver errors can flip the sign. Report the\n", "initial energy margin, uncertainty, observation duration, maximum radius, and whether\n", "radial velocity is outward. A growing radius over a short interval alone does not prove\n", "escape; a bound ellipse also moves outward for part of its orbit.\n", "\n", "Test the implementation with zero speed, circular speed, escape speed, and a clearly\n", "super-escape speed. Reversing a tangential velocity should preserve energy and reverse\n", "angular momentum. Changing only test-body mass should not change specific energy or the\n", "escape threshold.\n", "\n", "The Newtonian result is reliable for weak gravitational fields and ordinary planetary\n", "speeds. Strong fields near compact objects require relativity, where coordinate choices\n", "and spacetime geometry replace this simple energy derivation.\n", "\n", "## A worked Python notebook\n", "\n", "Find the speed needed to escape an isolated Earth-like spherical mass from a chosen radius. The question for this calculation is precise: how does **escape speed**\n", "change when we change **launch radius**, 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", "Atmospheric drag, thrust after launch, rotation, and other gravitating bodies are omitted. 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", "6.371e+06 to 2e+07 m. It gives us a specific domain to discuss\n", "instead of leaving the graph's range to a software default.\n", "\n", "Use the notebook link above the lesson, run the setup cell, and then run the cells in\n", "order. The code below is also included there. NumPy handles the arrays of numerical\n", "values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these\n", "libraries or notebook cells are unfamiliar, start with\n", "[Your first physics notebook](https://tirta.co/a-byte-of-physics/your-first-physics-notebook).\n", "\n", "### State the relationship before choosing a graph\n", "\n", "In this small example, `q` stands for launch radius, expressed in m. The output is\n", "escape speed, expressed in 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", "$$\n", "f(q) = 28234746.0339207 \\sqrt{\\frac{1}{q}}\n", "$$\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 1e+07 m, the expression gives 8928.61 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(\"sqrt(2*3.986004418e14/q)\", locals={\"q\": q})\n", "reference_input = 10000000.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"m\")\n", "print(\"Predicted escape speed:\", reference_output, \"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", "11186.1 m/s. The midpoint, at 1.31855e+07 m, gives\n", "7775.63 m/s. The upper endpoint gives 6313.48 m/s.\n", "Read these together with the assumptions: they describe different choices of launch radius\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([6371000.0, 13185500.0, 20000000.0])\n", "outputs = np.asarray(evaluate(inputs), dtype=float)\n", "outputs = np.broadcast_to(outputs, inputs.shape)\n", "for input_value, output_value in zip(inputs, outputs):\n", " print(f\"{input_value:.6g} m -> {output_value:.6g} 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 launch radius; the vertical axis shows escape speed. 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(6371000.0, 20000000.0, 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(\"Launch radius (m)\")\n", "ax.set_ylabel(\"Escape speed (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", "escape speed change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 1e+07 m is\n", "-0.000446431 in those ratio units. Locally, the output decreases as the\n", "input increases. This statement is local; the sign or size of the slope can be different\n", "elsewhere on the curve.\n", "\n", "\n", "$$\n", "f'(q) = - \\frac{14117373.0169603 \\sqrt{\\frac{1}{q}}}{q}\n", "$$\n", "\n", "\n", "For a small input change, multiply the derivative by that change to estimate the output\n", "change. Then compare that estimate with a fresh evaluation of the full expression.\n", "The distinction between these two calculations matters: the derivative gives a tangent\n", "approximation, while substitution evaluates the chosen expression at the new point." ] }, { "cell_type": "code", "id": "code-015", "metadata": {}, "source": [ "slope_expression = sp.diff(expression, q)\n", "slope_at_reference = float(slope_expression.subs(q, reference_input))\n", "changed_input = 10068145.0\n", "linear_prediction = reference_output + slope_at_reference * (\n", " changed_input - reference_input\n", ")\n", "full_prediction = float(evaluate(changed_input))\n", "print(\"Symbolic derivative:\", slope_expression)\n", "print(\"Tangent prediction:\", linear_prediction, \"m/s\")\n", "print(\"Full expression:\", full_prediction, \"m/s\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m/s\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 1.00681e+07 m. The tangent estimate is\n", "8898.19 m/s, while the full expression gives\n", "8898.34 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 = 13629000.0\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", "launch radius inside the stated interval, predict whether escape speed 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: Atmospheric drag, thrust after launch, rotation, and other gravitating bodies are omitted. 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 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(20000000.0) - evaluate(6371000.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(6371000.0, 20000000.0, 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 13.3, \"Gravitational Potential Energy and Total Energy\"](https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy).\n", "- [OpenStax, *University Physics Volume 1*, Section 13.4, \"Satellite Orbits and Energy\"](https://openstax.org/books/university-physics-volume-1/pages/13-4-satellite-orbits-and-energy).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 25, Sections 25.3-25.4](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter25new/).\n", "- [Benjamin Crowell, *Light and Matter*, gravitational energy and escape discussions](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Escape Velocity\"](https://physics.info/gravitation-escape/).\n", "- [Physics LibreTexts, \"Escape Velocity\"](https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_%28OpenStax%29/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_%28OpenStax%29/13%3A_Gravitation/13.04%3A_Gravitational_Potential_Energy_and_Total_Energy)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "escape-velocity-energy.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/escape-velocity-energy", "source_sha256": "8c0ed3be48aea5f61657f36f87fb1619cd980e91eb82c446dfe0df407851331d" } }, "nbformat": 4, "nbformat_minor": 5 }