A Byte of Physics

From equations to code

An equation and a line of code can look similar while meaning different things. An equation states that two expressions represent the same physical quantity. An assignment evaluates its right side now and stores the result on its left. Translating safely means understanding the relationship first, choosing which quantity is unknown, and only then choosing an order of operations.

Start with the physical statement

Consider one-dimensional motion at constant velocity. Choose an origin, choose the positive direction, and measure time from an agreed starting event. Position is then

\[x(t)=x_0+v(t-t_0)\]

The symbols mean:

• \(x(t)\) is the position at the requested time.
• \(x_0\) is the position at the reference time.
• \(v\) is constant throughout the interval and can be positive, zero, or negative.
• \(t-t_0\) is the elapsed time, not necessarily the clock reading itself.

All positions in one coordinate system must use the same length unit. Velocity has dimensions of length divided by time. Therefore the displacement term has dimensions

\[[v(t-t_0)]=\frac{L}{T}T=L\]

and it can be added to the initial position. This unit check would reject a common translation error such as adding velocity directly to position.

When the reference time is zero, the relation becomes the familiar form

\[x(t)=x_0+vt\]

The shorter equation has one extra convention: \(t_0=0\). That convention should be stated instead of being hidden in code.

One relationship, several possible programs

If the question asks for position, solve for position. If it asks when an object reaches a known position, rearrange the same relationship:

\[t=t_0+\frac{x-x_0}{v},\qquad v\ne0\]

If it asks for velocity, rearrange again:

\[v=\frac{x-x_0}{t-t_0},\qquad t\ne t_0\]

The original equality did not declare one variable to be the output. The problem did. Each rearrangement also reveals a different invalid input: computing arrival time requires nonzero velocity, while computing velocity requires a nonzero time interval.

The position calculation can be written vertically before it is compressed into a function:

initial_position_metres = 2.0
velocity_metres_per_second = -3.0
initial_time_seconds = 1.0
requested_time_seconds = 5.0
elapsed_time_seconds = requested_time_seconds - initial_time_seconds
displacement_metres = velocity_metres_per_second * elapsed_time_seconds
position_metres = initial_position_metres + displacement_metres
print(f"Position: {position_metres:.1f} m")

from dataclasses import dataclass, replace


@dataclass(frozen=True)
class ConstantVelocityMotion:
    initial_position_metres: float
    velocity_metres_per_second: float
    initial_time_seconds: float


def position_at(
    motion: ConstantVelocityMotion,
    requested_time_seconds: float,
) -> float:
    elapsed_time_seconds = requested_time_seconds - motion.initial_time_seconds
    displacement_metres = motion.velocity_metres_per_second * elapsed_time_seconds

    return motion.initial_position_metres + displacement_metres

Long names are useful here. Python's basic float type cannot distinguish metres from seconds, so the names carry information the type system does not. A unit-aware library can make that constraint stronger, but it cannot decide whether the physical model is appropriate.

Work one case by hand

Let the initial position be \(x_0=2\,\mathrm{m}\) at \(t_0=1\,\mathrm{s}\). Let the constant velocity be \(v=-3\,\mathrm{m/s}\). At \(t=5\,\mathrm{s}\), the elapsed time is \(4\,\mathrm{s}\) and

\[x=2\,\mathrm{m}+(-3\,\mathrm{m/s})(4\,\mathrm{s})=-10\,\mathrm{m}\]

The negative result is not a negative distance travelled. It is a position ten metres on the negative side of the chosen origin. The object travelled twelve metres during the interval. This distinction between position, displacement, and distance is physical meaning that a bare arithmetic expression does not preserve for you.

Closed form or update loop

For constant velocity, calling position_at for each requested time evaluates the exact model up to floating-point rounding. A time-stepping loop is unnecessary. Still, seeing the update form is useful because later models will not always have a convenient closed form:

\[x_{n+1}=x_n+v\,\Delta t\]

position_metres = 2.0
time_seconds = 1.0
velocity_metres_per_second = -3.0
end_time_seconds = 5.0
time_step_seconds = 0.25

while time_seconds < end_time_seconds:
    step_seconds = min(time_step_seconds, end_time_seconds - time_seconds)
    position_metres += velocity_metres_per_second * step_seconds
    time_seconds += step_seconds

print(f"At {time_seconds:.2f} s: {position_metres:.2f} m")

With a constant velocity and a final time that is an exact number of steps away, this update represents the same linear relationship. With changing velocity, the choice of which velocity to use during a step becomes a numerical approximation. That is where an algorithm begins to add error beyond ordinary floating-point arithmetic.

Checks that follow from the equation

Tests should come from physical or mathematical consequences, not arbitrary example outputs.

motion = ConstantVelocityMotion(
    initial_position_metres=2,
    velocity_metres_per_second=-3,
    initial_time_seconds=1,
)

assert position_at(motion, 1) == 2

stationary_motion = replace(
    motion,
    velocity_metres_per_second=0,
)

assert position_at(stationary_motion, 100) == 2

Other useful checks are dimensional consistency and linear scaling. At fixed velocity, doubling elapsed time doubles displacement, not necessarily position. Reversing the sign of velocity reverses the displacement. Changing the coordinate origin changes both position values by the same offset but does not change the displacement.

Know the preconditions

The relation assumes motion along one chosen axis and a velocity that remains constant over the interval. It does not describe a turning object, an accelerating object, or a coordinate system that moves during the calculation without additional definitions. It also says nothing about why the object has that velocity; dynamics enters when forces and acceleration are considered.

Good translation therefore has four visible layers: the physical assumptions, the equation, the algebraic form chosen for the question, and the implementation. If the output is surprising, checking those layers separately is much faster than staring at the final line of code.

Algebra comes before substitution

Keeping symbols until the relationship has been rearranged reduces both arithmetic and conceptual mistakes. Suppose an object begins at a known position and must reach a target. Solving symbolically for arrival time shows immediately that the denominator is velocity and that zero velocity is a special case. Substituting numbers too early can hide that division behind a calculator error or, worse, an infinity that code later treats as ordinary data.

The rearranged form also makes sensitivity visible. When the speed is small, a small change in speed can produce a large change in predicted arrival time. When the target and initial positions are almost equal, measurement uncertainty in either position can be comparable with the displacement itself. These facts follow from the structure of the expression before a program runs.

After algebra, write a worked example with units on every substituted quantity. Then translate it to code using one intermediate value for each physical idea. This makes a debugging trace readable: elapsed time, displacement, and final position can be checked separately. A single compressed expression may return the same number, but it gives the reviewer fewer places to compare the implementation with the equation.

Finally, keep the sign convention in the API documentation or nearby prose. A negative velocity is not an error when the positive axis points the other way. Rejecting every negative input would encode a false physical restriction. Validation should reject quantities that violate the model's domain, not values that merely describe motion in a different allowed direction.

A worked Python notebook

A cart starts at 2 metres and moves at a constant velocity of minus 3 metres per second. The question for this calculation is precise: how does position change when we change elapsed time, 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.

The cart must remain on a straight track and keep the same velocity. 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 0 to 4 s. 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 elapsed time, expressed in s. The output is position, expressed in m. 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.

\[f(q) = 2 - 3 q\]

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 2 s, the expression gives -4 m. 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("2-3*q", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted position:", reference_output, "m")

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 2 m. The midpoint, at 2 s, gives -4 m. The upper endpoint gives -10 m. Read these together with the assumptions: they describe different choices of elapsed time 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([0.0, 2.0, 4.0])
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} s -> {output_value:.6g} m")
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 elapsed time; the vertical axis shows position. 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(0.0, 4.0, 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("Elapsed time (s)")
ax.set_ylabel("Position (m)")
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 position change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 s is -3 in those ratio units. Locally, the output decreases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = -3\]

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 = 2.02
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, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 2.02 s. The tangent estimate is -4.06 m, while the full expression gives -4.06 m. 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 = 4.0
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 elapsed time inside the stated interval, predict whether position 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: The cart must remain on a straight track and keep the same velocity. 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 m: 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(4.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 4.0, 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.

Sources

• OpenStax, University Physics Volume 1, Section 3.2, "Instantaneous Velocity and Speed".
• OpenStax, University Physics Volume 1, Section 3.4, "Motion with Constant Acceleration".
• Benjamin Crowell, Light and Matter, Chapter 1, "Scaling and Order-of-Magnitude Estimates".
• MIT OpenCourseWare, 8.01SC Classical Mechanics, Chapter 4, "One Dimensional Kinematics".