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Kinetic energy from data

Translational kinetic energy turns mass and velocity into a scalar measure of energy of motion. Direction disappears from the final value, but the original velocity vector, reference frame, time, units, and measurement method remain essential. A speed copied from a rounded display is weaker evidence than the underlying timestamped state.

Classical translational kinetic energy

For a particle or body whose center of mass moves at speed \(v\) in a stated inertial frame,

\[K=\frac12mv^2=\frac12m\mathbf v\cdot\mathbf v\]

Mass must be nonnegative in this ordinary model, and velocity components must share one basis and unit. Because a vector dotted with itself is nonnegative, classical kinetic energy is nonnegative. It is zero when velocity is zero in the selected frame.

In SI base units,

\[[K]=\mathrm{kg\,m^2/s^2}=\mathrm J\]

Reversing velocity leaves energy unchanged: \(K(-\mathbf v)=K(\mathbf v)\). Momentum reverses, so kinetic energy alone cannot reconstruct the state.

import numpy as np


def kinetic_energy(mass, velocity):
    velocity = np.asarray(velocity, dtype=float)
    if mass < 0 or not np.all(np.isfinite(velocity)):
        raise ValueError("Use nonnegative mass and finite velocity.")
    return 0.5 * mass * np.dot(velocity, velocity)


print(f"Energy: {kinetic_energy(2.0, [3.0, 4.0]):.1f} J")

For ordinary measured matter, negative mass input is invalid. A zero mass with finite classical velocity produces zero in the algebra but does not model a photon; relativistic massless particles require a different energy relation.

Scaling predictions

At fixed speed, doubling mass doubles kinetic energy. At fixed mass, doubling speed quadruples it:

\[K(cm,v)=cK(m,v),\qquad K(m,cv)=c^2K(m,v)\]

These relations are useful checks because they do not depend on a particular numeric example. They also explain why small speed errors can become more important after squaring.

For mass \(m=2.0\,\mathrm{kg}\) and velocity \(\mathbf v=(3,4)\,\mathrm{m/s}\), speed is \(5\,\mathrm{m/s}\) and

\[K=\frac12(2.0)(5^2)=25\,\mathrm J\]

Kinetic energy depends on reference frame

If another inertial frame moves at constant velocity \(\mathbf V\) relative to the first, classical velocity transforms as \(\mathbf v'=\mathbf v-\mathbf V\). Therefore

\[K'=\frac12m|\mathbf v-\mathbf V|^2\]

Expanding,

\[K'=K-m\mathbf V\cdot\mathbf v+\frac12mV^2\]

Kinetic energy is not Galilean invariant. A passenger at rest in a train has zero kinetic energy relative to the train but nonzero kinetic energy relative to the ground. This is not a contradiction: energy accounting, work, and system boundaries must use a consistent frame.

measured_velocity = np.array([3.0, 4.0])
frame_velocity = np.array([3.0, 0.0])
relative_velocity = measured_velocity - frame_velocity
print(f"Energy in moving frame: {kinetic_energy(2.0, relative_velocity):.1f} J")

The frame velocity must be expressed in the same basis, units, and timestamp convention as the measured velocity.

A system of particles

For particles labeled by \(i\),

\[K_{\mathrm{total}}=\sum_i\frac12m_i|\mathbf v_i|^2\]

Let total mass be \(M=\sum_i m_i\) and center-of-mass velocity

\[\mathbf V_{\mathrm{CM}}=\frac{1}{M}\sum_i m_i\mathbf v_i\]

Writing each velocity as center-of-mass velocity plus a relative part gives

\[K_{\mathrm{total}}=\frac12M V_{\mathrm{CM}}^2+\sum_i\frac12m_i|\mathbf v_i-\mathbf V_{\mathrm{CM}}|^2\]

The first term is bulk translational energy. The second is kinetic energy relative to the center of mass. For a gas, that relative microscopic motion contributes to internal energy, but internal energy also depends on molecular rotation, vibration, and interactions. It should not be identified with one unspecific kinetic-energy sum.

masses = np.array([1.0, 2.0])
velocities = np.array([[3.0, 0.0], [-1.0, 0.0]])
com_velocity = np.sum(masses[:, None] * velocities, axis=0) / masses.sum()
bulk_energy = kinetic_energy(masses.sum(), com_velocity)
relative_energy = sum(
    kinetic_energy(m, v - com_velocity) for m, v in zip(masses, velocities)
)
total = sum(kinetic_energy(m, v) for m, v in zip(masses, velocities))
assert np.isclose(total, bulk_energy + relative_energy)
print("Total, bulk, relative energies (J):", total, bulk_energy, relative_energy)

The two terms should add back to the direct particle sum within floating-point tolerance.

Rotation is another energy store

For a rigid body, center-of-mass translation is not the whole kinetic energy. Rotation about the center of mass contributes

\[K_{\mathrm{rot}}=\frac12I\omega^2\]

for a simple fixed-axis case. The full rigid-body relation uses the inertia tensor and angular-velocity vector. A rolling object can have both translational and rotational kinetic energy. Applying only

\(\tfrac12Mv_{\mathrm{CM}}^2\)

would omit the rotation.

Deformation and vibration can add further kinetic terms. The representation must match the system's modeled degrees of freedom.

Deriving velocity from position samples

Position data does not directly contain instantaneous velocity. For equally spaced samples, an interior centered estimate is

\[\mathbf v(t_i)\approx\frac{\mathbf r(t_{i+1})-\mathbf r(t_{i-1})}{t_{i+1}-t_{i-1}}\]

Use actual timestamps when spacing is uneven. A forward difference assigns an interval average to an endpoint unless a more careful time convention is stated. Differentiation amplifies high-frequency measurement noise, and squaring speed introduces positive bias: zero-mean velocity noise does not generally average to zero energy error.

times = np.linspace(0, 2, 21)
positions = 3 * times + 0.5 * times**2
velocities = np.gradient(positions, times, edge_order=2)
energies = 0.5 * 2.0 * velocities**2
print("Energy at 1 s (J):", energies[10])

Filtering or curve fitting can reduce noise but adds assumptions and may erase rapid motion. Report the derivative or fit method, window, boundary handling, and sampling rate.

Uncertainty propagation

For independent small uncertainties in scalar mass and speed,

\[\sigma_K^2\approx\left(\frac12v^2\sigma_m\right)^2+(mv\sigma_v)^2\]

If mass and speed estimates are correlated, add the covariance term. For vector velocity with covariance matrix \(\boldsymbol\Sigma_v\), linear propagation away from zero velocity uses gradient \(\partial K/\partial\mathbf v=m\mathbf v\).

Near zero velocity, first-order propagation can be misleading because that gradient vanishes while squared noise still produces positive estimated energy. Monte Carlo propagation or a measurement-specific statistical model can be more appropriate.

Work-energy check

For a constant-mass particle,

\[\Delta K=W_{\mathrm{net}}=\int\mathbf F_{\mathrm{net}}\cdot d\mathbf r\]

Compute the kinetic-energy difference from endpoint velocity data and net work from force-position data independently. Agreement checks timestamps, force signs, units, and path integration. If the data describe an extended or changing-mass system, clarify which work-energy statement applies before treating disagreement as error.

Relativistic limit

Classical kinetic energy is accurate when speed is much smaller than the speed of light. Special relativity uses

\[K=(\gamma-1)mc^2,\qquad \gamma=\frac{1}{\sqrt{1-v^2/c^2}}\]

For \(v\ll c\), its series begins with \(\tfrac12mv^2\). The classical formula increasingly underestimates energy as speed approaches \(c\) and cannot describe a massive body at or above that speed.

Validation checks

Kinetic energy is easy to calculate only after the data's meaning is secure. Mass, velocity, frame, timestamp, modeled degrees of freedom, and uncertainty determine what the scalar result actually represents.

A worked Python notebook

Compare kinetic energies for a 2 kilogram body moving at different speeds. The question for this calculation is precise: how does kinetic energy change when we change speed, 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.

Classical kinetic energy applies when speeds are small compared with light speed. Keep that sentence beside the results. A formula can return a number outside 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 8 m/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 speed, expressed in m/s. The output is kinetic energy, expressed in J. 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.

\[f(q) = q^{2}\]

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 5 m/s, the expression gives 25 J. 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("q**2", locals={"q": q})
reference_input = 5.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "m/s")
print("Predicted kinetic energy:", reference_output, "J")

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 J. The midpoint, at 4 m/s, gives 16 J. The upper endpoint gives 64 J. Read these together with the assumptions: they describe different choices of speed 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, 4.0, 8.0])
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/s -> {output_value:.6g} J")
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 speed; the vertical axis shows kinetic energy. 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, 8.0, 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("Speed (m/s)")
ax.set_ylabel("Kinetic energy (J)")
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 kinetic energy change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 5 m/s is 10 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.

\[f'(q) = 2 q\]

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 = 5.04
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, "J")
print("Full expression:", full_prediction, "J")
print("Approximation difference:", full_prediction - linear_prediction, "J")

Here the changed input is 5.04 m/s. The tangent estimate is 25.4 J, while the full expression gives 25.4016 J. 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 = 8.0
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 speed inside the stated interval, predict whether kinetic energy 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: Classical kinetic energy applies when speeds are small compared with light speed. 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 J: 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(8.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 8.0, 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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