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Physics as Modeling

Physics is the practice of building limited descriptions of nature, using them to make predictions, and checking those predictions against observations or trusted results. An equation is one part of that process. It becomes a useful model only after we identify the system, choose what to include, define the quantities, and state the conditions under which the relationship is expected to work.

A model is not defective merely because it leaves something out. Every useful model is selective. A constant-acceleration model can describe a falling object over a modest height without tracking individual air molecules or Earth's orbit. Its value comes from retaining the effects that matter for the question while discarding effects too small to change the answer at the required accuracy.

Begin with a physical question

“Model a projectile” is too vague. “Estimate where a slowly thrown ball lands, assuming level ground and negligible air resistance” specifies an output and suggests a scope. Before calculating, write down:

These choices prevent hidden changes of meaning. For example, work done on a system and work done by a system use opposite common sign conventions. Potential energy depends on a chosen zero. Velocity depends on a reference frame. Two correct calculations can look inconsistent when these definitions are not aligned.

Idealization is a controlled decision

Suppose an object of mass \(m\) moves near Earth's surface. A first model might keep only a constant gravitational force:

\[\mathbf{F}_{\mathrm{net}}=m\mathbf{a}=m\mathbf{g}\]

Cancelling a nonzero mass gives \(\mathbf{a}=\mathbf{g}\). This does not mean all falling objects behave identically in every circumstance. It means mass cancels within a model that has omitted drag, buoyancy, changing gravity, rotation of Earth, and other interactions.

The same model may be excellent for one question and poor for another. Neglecting air resistance might be adequate for a dense ball travelling a short distance, but not for a sheet of paper or a long-range projectile. “Negligible” must mean small relative to the required accuracy, not simply inconvenient to calculate.

From principles to a calculable model

The path from a physical idea to a result has several layers:

  1. Conceptual model: identify objects, interactions, and approximations.
  2. Mathematical model: express those choices with quantities and relationships.
  3. Numerical representation: choose finite precision, time steps, grids, or samples when an analytic solution is unavailable or inconvenient.
  4. Implementation: translate the numerical method into code.
  5. Comparison: test the implementation and the physical prediction.

Keeping the layers separate improves diagnosis. A test may show that code solves the stated differential equation correctly, while experimental data show that the equation omits an important force. Conversely, a sound physical equation can give wrong output because its implementation mixes units or updates state incorrectly.

import numpy as np

height_m, gravity = 10.0, 9.81
landing_time = np.sqrt(2 * height_m / gravity)
assert np.isclose(0.5 * gravity * landing_time**2, height_m)
print(f"Fall from rest without drag: {landing_time:.3f} s")

Verification asks, “Did we solve the equations as intended?” Validation asks, “Do these equations describe the real situation well enough for this purpose?” The two questions are related but not interchangeable.

Dimensional reasoning before arithmetic

Units are part of a physical quantity, not decoration added after a calculation. If a proposed travel distance is

\[x=v_0t+\frac{1}{2}at^2\]

then both terms on the right must have dimensions of length. \(v_0t\) has units of speed times time, and \(at^2\) also has units of length. A sum of seconds and metres would signal an error even if a calculator returned a number.

Dimensionless arguments deserve special care. The input to an exponential, logarithm, or trigonometric function must be dimensionless. Unit checking cannot prove an equation correct, because a wrong dimensionless factor can remain, but it eliminates many impossible formulas quickly.

Scale, approximation, and nondimensional numbers

Models often depend on the relative size of effects rather than on any single quantity. A small parameter

\[\epsilon=\frac{\text{neglected effect}}{\text{retained effect}}\]

supports an approximation when

\(|\epsilon|\ll 1\)

throughout the relevant range. This is stronger than saying an effect “looks small.” It states what the effect is compared with and invites a quantitative check.

Characteristic scales can also convert variables into dimensionless form. If \(L\) and \(T\) are typical scales, then \(x/L\) and \(t/T\) reveal which parameter combinations control the behaviour. Systems with very different physical sizes may follow the same dimensionless equation.

Analytic and numerical models support each other

An analytic expression exposes dependence on parameters and can provide exact limits. A numerical method handles equations that cannot be solved conveniently in closed form. Neither approach automatically outranks the other. A good workflow uses simple analytic cases to test numerical code, then uses the numerical method where the analytic simplification no longer applies.

For a state \(y(t)\) satisfying \(dy/dt=f(t,y)\), an explicit Euler update is

\[y_{n+1}=y_n+\Delta t\,f(t_n,y_n)\]

In plain steps:

distance, velocity, dt, duration = 0.0, 0.0, 0.001, 1.0
for _ in range(round(duration / dt)):
    distance += dt * velocity
    velocity += dt * gravity
print("Euler fall distance (m):", distance)
print("Exact distance (m):", 0.5 * gravity * duration**2)

The update is a numerical approximation, not a new law of physics. Its error depends on the method and step size. Repeating the same calculation with smaller steps tests convergence. If results do not settle within the required tolerance, the discretization has not yet earned trust.

Checks that reveal different mistakes

No single test establishes correctness. Combine checks with different purposes:

Passing a conservation test is evidence only when the modeled system should conserve that quantity. Mechanical energy need not remain constant when the system definition permits energy transfer by friction. A test must match the model's boundary and assumptions.

Measurement and uncertainty

Measured inputs are not exact real numbers. A reported measurement can be written as a central value and uncertainty, such as \(x\pm u_x\). Uncertainty describes a justified range or distribution; it is not the same as a mistake. Repeated random variation, calibration bias, resolution, and model assumptions can contribute in different ways.

For a calculated quantity \(q=f(x_1,x_2,\ldots)\), input uncertainties can be propagated analytically for simple cases or sampled numerically for more complicated ones. Report numerical error, measurement uncertainty, and model inadequacy separately when possible. A result can have tiny rounding error and still be physically uncertain.

rng = np.random.default_rng(42)
heights = rng.normal(10.0, 0.1, 10000)
times = np.sqrt(2 * heights / gravity)
print("Mean fall time (s):", times.mean())
print("Central 95% input-propagated interval (s):", np.quantile(times, [0.025, 0.975]))

Sampling requires enough draws, an appropriate distribution model, and a recorded random seed when exact reproduction matters. More samples reduce sampling noise; they do not correct biased measurements or a missing physical interaction.

Refinement should answer a demonstrated need

Add complexity only after a comparison shows why it matters. If a vacuum projectile model misses observed range systematically, introduce a drag law and test again. If constant gravity is inadequate over a large altitude, replace it with a distance-dependent field. Each refinement should name the failed assumption, the new parameter or state, and the evidence that supports the change.

More detailed models cost more to understand, compute, and validate. They can also be less reliable if they introduce poorly known parameters. The best model is not the one with the most mechanisms; it is the simplest one whose errors are small enough for the stated question.

Communicating a model honestly

A complete result includes the predicted quantity, units, uncertainty or tolerance, assumptions, validity range, and comparison used to check it. Graphs must label axes and distinguish measured points from calculated curves. Tables should retain enough digits for reproducibility without implying unsupported precision. Code, parameter files, and data versions should be recorded when the result depends on them.

The final habit is to make the claim no broader than the evidence. A classroom model can illuminate a principle without being suitable for engineering, medical, environmental, or safety decisions. Stating limits does not weaken the model. It defines exactly what has been learned and makes the next refinement possible.

A worked Python notebook

Turn the question of landing time into a model with explicit height, initial rest, and gravity assumptions. The question for this calculation is precise: how does predicted fall time change when we change release height, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Comparing a prediction to observations is separate from checking its arithmetic. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 1 to 30 m. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for release height, expressed in m. The output is predicted fall time, expressed in s. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 0.451523640985731 \sqrt{q}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 10 m, the expression gives 1.42784 s. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("sqrt(2*q/9.81)", locals={"q": q})
reference_input = 10.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m")
print("Predicted predicted fall time:", reference_output, "s")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0.451524 s. The midpoint, at 15.5 m, gives 1.77765 s. The upper endpoint gives 2.4731 s. Read these together with the assumptions: they describe different choices of release height within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([1.0, 15.5, 30.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} m -> {output_value:.6g} s")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows release height; the vertical axis shows predicted fall time. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(1.0, 30.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Release height (m)")
ax.set_ylabel("Predicted fall time (s)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does predicted fall time change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 10 m is 0.0713922 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = \frac{0.225761820492865}{\sqrt{q}}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 10.145
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "s")
print("Full expression:", full_prediction, "s")
print("Approximation difference:", full_prediction - linear_prediction, "s")

Here the changed input is 10.145 m. The tangent estimate is 1.43819 s, while the full expression gives 1.43816 s. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 29.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of release height inside the stated interval, predict whether predicted fall time will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Comparing a prediction to observations is separate from checking its arithmetic. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of s: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(30.0) - evaluate(1.0))
for points in [101, 201, 401]:
    grid = np.linspace(1.0, 30.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

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