{ "cells": [ { "cell_type": "markdown", "id": "markdown-000", "metadata": {}, "source": [ "# Relative motion\n", "\n", "[Read the lesson](https://tirta.co/a-byte-of-physics/relative-motion). 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": [ "Position and velocity are measured relative to a reference frame. A passenger can be at\n", "rest relative to a train while moving rapidly relative to the ground. Relative-motion\n", "equations translate between these descriptions without declaring one observer's numbers\n", "to be the only real ones.\n", "\n", "## Read the subscripts as a sentence\n", "\n", "Let $\\mathbf r_{A/B}$ mean\n", "“the position of A relative to B.” If A and B are represented in a common coordinate\n", "frame with parallel axes, then\n", "\n", "\n", "$$\n", "\\mathbf r_{A/B}=\\mathbf r_A-\\mathbf r_B\n", "$$\n", "\n", "\n", "Reversing the order reverses the vector:\n", "\n", "\n", "$$\n", "\\mathbf r_{B/A}=-\\mathbf r_{A/B}\n", "$$\n", "\n", "\n", "This subtraction has a geometric meaning: it is the displacement from B to A. The\n", "vectors on the right must refer to the same origin, basis, units, and time. Subtracting\n", "components from differently oriented axes without first transforming the basis does not\n", "produce a valid relative position.\n", "\n", "## Relative velocity\n", "\n", "Differentiate the relative position with respect to a shared time:\n", "\n", "\n", "$$\n", "\\mathbf v_{A/B}=\\frac{d\\mathbf r_{A/B}}{dt}=\\mathbf v_A-\\mathbf v_B\n", "$$\n", "\n", "\n", "An equivalent chaining rule is\n", "\n", "\n", "$$\n", "\\mathbf v_{A/C}=\\mathbf v_{A/B}+\\mathbf v_{B/C}\n", "$$\n", "\n", "\n", "The middle labels cancel conceptually. This notation is much safer than trying to\n", "remember whether a particular word problem requires addition or subtraction." ] }, { "cell_type": "code", "id": "code-003", "metadata": {}, "source": [ "import numpy as np\n", "\n", "object_position = np.array([5.0, 3.0])\n", "observer_position = np.array([2.0, 1.0])\n", "object_velocity = np.array([4.0, 0.0])\n", "observer_velocity = np.array([1.0, 0.0])\n", "# All values refer to the same time and east/north laboratory axes.\n", "print(\"Relative position (m):\", object_position - observer_position)\n", "print(\"Relative velocity (m/s):\", object_velocity - observer_velocity)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-004", "metadata": {}, "source": [ "The identity labels are not decoration. They prevent a later calculation from silently\n", "reversing the ordered relationship.\n", "\n", "## Worked example: walking on a train\n", "\n", "Suppose a train moves east relative to the ground at\n", "$20\\,\\mathrm{m/s}$.\n", "A passenger walks toward the front at\n", "$1.5\\,\\mathrm{m/s}$\n", "relative to the train. With east positive,\n", "\n", "\n", "$$\n", "v_{P/G}=v_{P/T}+v_{T/G}=1.5+20=21.5\\,\\mathrm{m/s}\n", "$$\n", "\n", "\n", "If the passenger walks toward the rear at the same walking speed, then\n", "$v_{P/T}=-1.5\\,\\mathrm{m/s}$\n", "and the ground speed becomes\n", "$18.5\\,\\mathrm{m/s}$\n", "east. “Toward the rear” does not automatically mean west relative to the ground; it\n", "specifies the passenger's velocity relative to the train.\n", "\n", "The train's velocity relative to the passenger is\n", "$v_{T/P}=-1.5\\,\\mathrm{m/s}$,\n", "not negative twenty. The observer named after the slash determines the answer.\n", "\n", "## Worked example: boat and current\n", "\n", "Let east be the first component and north the second. A boat moves relative to the water\n", "at\n", "$\\mathbf v_{B/W}=(0,4)\\,\\mathrm{m/s}$.\n", "The water moves relative to the ground at\n", "$\\mathbf v_{W/G}=(3,0)\\,\\mathrm{m/s}$.\n", "Then\n", "\n", "\n", "$$\n", "\\mathbf v_{B/G}=\\mathbf v_{B/W}+\\mathbf v_{W/G}=(3,4)\\,\\mathrm{m/s}\n", "$$\n", "\n", "\n", "The ground speed is\n", "$5\\,\\mathrm{m/s}$ and the\n", "direction is northeast. Pointing the boat due north relative to the water does not make\n", "its path due north relative to shore.\n", "\n", "If the goal is to travel due north relative to ground, the boat must aim westward enough\n", "that its westward water-relative component cancels the eastward current. This is an\n", "inverse problem: specify the desired ground velocity, subtract the current, and check\n", "whether the required boat-relative speed is physically available." ] }, { "cell_type": "code", "id": "code-005", "metadata": {}, "source": [ "desired_ground_velocity = np.array([0.0, 3.0])\n", "water_velocity = np.array([1.0, 0.0])\n", "required_boat_velocity = desired_ground_velocity - water_velocity\n", "speed_limit = 4.0\n", "required_speed = np.linalg.norm(required_boat_velocity)\n", "print(\"Boat velocity relative to water (m/s):\", required_boat_velocity)\n", "print(\"Course is reachable:\", required_speed <= speed_limit)\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-006", "metadata": {}, "source": [ "## Galilean coordinate transformation\n", "\n", "Consider frame S prime moving with constant velocity\n", "$\\mathbf V$ relative to frame S.\n", "If their origins coincide at time zero and their axes remain parallel, classical\n", "mechanics uses\n", "\n", "\n", "$$\n", "\\mathbf r'=\\mathbf r-\\mathbf Vt,\\qquad t'=t\n", "$$\n", "\n", "\n", "Differentiating gives\n", "\n", "\n", "$$\n", "\\mathbf v'=\\mathbf v-\\mathbf V\n", "$$\n", "\n", "\n", "and, when $\\mathbf V$ is\n", "constant,\n", "\n", "\n", "$$\n", "\\mathbf a'=\\mathbf a\n", "$$\n", "\n", "\n", "This shared acceleration is why Newton's second law keeps the same form in inertial\n", "frames under Galilean transformations. It does not mean velocity or kinetic energy is\n", "frame-independent.\n", "\n", "If the frame origins do not coincide at time zero, include an initial offset. If the\n", "axes are rotated, transform components as well. A translation-only formula should not\n", "silently pretend to handle every frame relationship.\n", "\n", "## Synchronizing data in time\n", "\n", "Relative position requires simultaneous states according to the model's time\n", "convention. Subtracting an object's position recorded at one time from an observer's\n", "position at another mixes spatial separation with time evolution.\n", "\n", "For sampled data, interpolate both trajectories to a common timestamp before\n", "subtracting. The interpolation rule becomes part of the result's uncertainty. A stale\n", "observer sample can produce a plausible but false relative velocity." ] }, { "cell_type": "code", "id": "code-007", "metadata": {}, "source": [ "object_times = np.array([0.0, 1.0, 2.0])\n", "object_positions = np.array([0.0, 4.0, 8.0])\n", "observer_times = np.array([0.0, 0.5, 1.5, 2.0])\n", "observer_positions = np.array([0.0, 0.5, 1.5, 2.0])\n", "requested_time = 1.0\n", "relative_position = np.interp(\n", " requested_time, object_times, object_positions\n", ") - np.interp(requested_time, observer_times, observer_positions)\n", "print(f\"Relative position at 1 s: {relative_position:.1f} m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-008", "metadata": {}, "source": [ "## Relative acceleration and noninertial frames\n", "\n", "For general translational motion,\n", "\n", "\n", "$$\n", "\\mathbf a_{A/B}=\\mathbf a_A-\\mathbf a_B\n", "$$\n", "\n", "\n", "If the observer frame accelerates, an object with zero acceleration in an inertial frame\n", "appears to accelerate in the observer frame. To write Newton's second law using only\n", "measurements in that accelerating frame, an inertial or fictitious force is introduced.\n", "For a frame translating with acceleration\n", "$\\mathbf A$,\n", "\n", "\n", "$$\n", "\\mathbf F_{\\mathrm{inertial}}=-m\\mathbf A\n", "$$\n", "\n", "\n", "Rotating frames add further terms, including centrifugal and Coriolis effects. Simple\n", "vector subtraction between parallel translating axes does not cover rotating bases;\n", "the time derivative of the basis vectors also matters.\n", "\n", "## Closing speed is not always relative speed\n", "\n", "Relative speed is\n", "$|\\mathbf v_{A/B}|$.\n", "Closing speed is the rate at which the separation distance decreases. Let\n", "$\\hat{\\mathbf n}$ point\n", "along the instantaneous separation. Then the separation rate is\n", "\n", "\n", "$$\n", "\\frac{d}{dt}|\\mathbf r_{A/B}|=\\mathbf v_{A/B}\\cdot\\hat{\\mathbf n}\n", "$$\n", "\n", "\n", "Closing speed is the negative of this rate when the distance is decreasing. Two objects\n", "can have a large relative speed while moving sideways with little instantaneous change\n", "in separation. Radar range rate and collision prediction depend on this distinction.\n", "\n", "## Classical limit and relativity\n", "\n", "Galilean velocity addition is an excellent approximation when all relevant speeds are\n", "small compared with the speed of light. At relativistic speeds, observers do not share\n", "an absolute time, and velocities do not simply add without bound. In one dimension the\n", "special-relativistic transformation is\n", "\n", "\n", "$$\n", "u'=\\frac{u-V}{1-uV/c^2}\n", "$$\n", "\n", "\n", "For speeds much smaller than\n", "$c$, the denominator is nearly one\n", "and the Galilean rule is recovered. The applicable regime must be stated rather than\n", "assuming classical subtraction for every speed.\n", "\n", "## Checks for relative-motion calculations\n", "\n", "- Reversing the object and observer negates relative position and velocity.\n", "- An object compared with itself has zero relative state.\n", "- Chaining through an intermediate frame gives the direct result.\n", "- All terms use the same basis, units, and timestamp.\n", "- Adding a common constant velocity to both objects leaves their relative velocity\n", " unchanged.\n", "- Constant-velocity Galilean frames agree on acceleration.\n", "- A stationary object in one frame appears with the negative frame velocity in another.\n", "- The low-speed relativistic formula approaches classical subtraction.\n", "- Closing speed is computed from the separation direction, not assumed equal to relative\n", " speed.\n", "- Frame translations, rotations, and accelerations are not conflated.\n", "\n", "Relative motion is systematic once every quantity names both its subject and its\n", "observer. The equations are short; most mistakes come from losing those labels,\n", "combining unsynchronized data, or using a transformation outside its assumptions.\n", "\n", "## A worked Python notebook\n", "\n", "An object moves at 4 metres per second while an observer moves at 1 metre per second along the same axis. The question for this calculation is precise: how does **relative position**\n", "change when we change **time**, 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 subtraction uses simultaneous observations in the same inertial frame. 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 5 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 time, expressed in s. The output is\n", "relative position, expressed in m. 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", "$$\n", "f(q) = 3 q\n", "$$\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 2 s, the expression gives 6 m. 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(\"3*q\", locals={\"q\": q})\n", "reference_input = 2.0\n", "reference_output = float(expression.subs(q, reference_input))\n", "print(\"Input:\", reference_input, \"s\")\n", "print(\"Predicted relative position:\", reference_output, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-010", "metadata": {}, "source": [ "`sp.Symbol` declares a mathematical symbol. It does not assign a numerical measurement\n", "to that symbol. `subs` then replaces the symbol with our chosen input, and `float`\n", "converts the resulting symbolic number into a standard Python numerical value.\n", "Keeping these steps separate lets us inspect the equation before numerical substitution.\n", "It also prevents a later plot from becoming the only surviving record of the model.\n", "\n", "### Compare a few cases you can explain\n", "\n", "Before drawing hundreds of points, calculate three. The lower endpoint gives\n", "0 m. The midpoint, at 2.5 s, gives\n", "7.5 m. The upper endpoint gives 15 m.\n", "Read these together with the assumptions: they describe different choices of time\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([0.0, 2.5, 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} s -> {output_value:.6g} m\")\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 time; the vertical axis shows relative position. 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(0.0, 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(\"Time (s)\")\n", "ax.set_ylabel(\"Relative position (m)\")\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", "relative position change near the reference input? Its units are the output unit divided by\n", "the input unit. For this calculation, the derivative at 2 s is\n", "3 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", "$$\n", "f'(q) = 3\n", "$$\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 = 2.025\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\")\n", "print(\"Full expression:\", full_prediction, \"m\")\n", "print(\"Approximation difference:\", full_prediction - linear_prediction, \"m\")\n" ], "execution_count": null, "outputs": [] }, { "cell_type": "markdown", "id": "markdown-016", "metadata": {}, "source": [ "Here the changed input is 2.025 s. The tangent estimate is\n", "6.075 m, while the full expression gives\n", "6.075 m. 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 = 5.0\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", "time inside the stated interval, predict whether relative position 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 subtraction uses simultaneous observations in the same inertial frame. 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:\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(0.0))\n", "for points in [101, 201, 401]:\n", " grid = np.linspace(0.0, 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 4.6, \"Relative Motion in One and Two Dimensions\"](https://openstax.org/books/university-physics-volume-1/pages/4-6-relative-motion-in-one-and-two-dimensions).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 4, \"One Dimensional Kinematics\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter4/).\n", "- [MIT OpenCourseWare, *8.01SC Classical Mechanics*, Chapter 5, \"Two Dimensional Kinematics\"](https://ocw.mit.edu/courses/8-01sc-classical-mechanics-fall-2016/resources/mit8_01scs22_chapter5/).\n", "- [Benjamin Crowell, *Light and Matter*, Chapter 8, \"Vectors and Motion\"](https://open.umn.edu/opentextbooks/textbooks/light-and-matter).\n", "- [The Physics Hypertextbook, \"Relative Velocity\"](https://physics.info/velocity-relative/)." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "name": "python", "version": "3.12" }, "colab": { "name": "relative-motion.ipynb", "provenance": [] }, "tirta": { "lesson": "https://tirta.co/a-byte-of-physics/relative-motion", "source_sha256": "adff329590533fe901255df0fa8e9122dec52fddfc6c62a76f1b82c4663800f4" } }, "nbformat": 4, "nbformat_minor": 5 }