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Parametric motion

Motion in more than one dimension is described by making every position component a function of the same time parameter. The resulting parametric curve tells where the object is at each instant, not just which geometric path it visits. Velocity and acceleration follow by differentiating the position vector with respect to time.

Position as a vector-valued function

In three-dimensional Cartesian coordinates,

\[\mathbf r(t)=x(t)\hat{\mathbf i}+y(t)\hat{\mathbf j}+z(t)\hat{\mathbf k}\]

The coordinate functions share one time variable. The geometric curve is the set of positions reached, while the parameterization also records order, timing, direction, and speed. Two formulas can trace the same curve with different motion.

For example,

\(\mathbf r_1(t)=(t,t^2)\)

and

\(\mathbf r_2(t)=(2t,4t^2)\)

both lie on \(y=x^2\), but at the same time their positions and velocities generally differ. The path equation alone does not specify the motion.

The parameter need not always be physical time. A curve may use an angle or arbitrary parameter. Calling derivatives velocity and acceleration is valid only when the parameter is time or has been converted to time through the chain rule.

Velocity is tangent to the path

Average velocity over a nonzero interval is

\[\mathbf v_{\mathrm{avg}}=\frac{\mathbf r(t+\Delta t)-\mathbf r(t)}{\Delta t}\]

The instantaneous velocity is the limit:

\[\mathbf v(t)=\frac{d\mathbf r}{dt}=\frac{dx}{dt}\hat{\mathbf i}+\frac{dy}{dt}\hat{\mathbf j}+\frac{dz}{dt}\hat{\mathbf k}\]

At a regular point where velocity is nonzero, the velocity vector is tangent to the trajectory and points in the direction of increasing time. Speed is its magnitude,

\[v(t)=|\mathbf v(t)|=\sqrt{v_x^2+v_y^2+v_z^2}\]

Speed is a scalar and is never negative. A component can be negative because it records direction along a chosen axis.

Acceleration changes velocity

Acceleration is

\[\mathbf a(t)=\frac{d\mathbf v}{dt}=\frac{d^2\mathbf r}{dt^2}\]

Acceleration can change speed, direction, or both. It does not have to point along the path. A car rounding a curve at constant speed has acceleration because its velocity direction changes.

The component equations are independent definitions in a fixed Cartesian basis:

\[v_x=\frac{dx}{dt},\quad v_y=\frac{dy}{dt},\quad a_x=\frac{dv_x}{dt},\quad a_y=\frac{dv_y}{dt}\]

They can be calculated separately, but their time values, units, frame, and basis must remain shared.

Worked example: uniform circular motion

Consider a circle of radius \(R\) centered at the origin, traversed counterclockwise with constant angular speed \(\omega\):

\[\mathbf r(t)=R\cos(\omega t)\hat{\mathbf i}+R\sin(\omega t)\hat{\mathbf j}\]

Differentiating gives

\[\mathbf v(t)=-R\omega\sin(\omega t)\hat{\mathbf i}+R\omega\cos(\omega t)\hat{\mathbf j}\]

and

\[\mathbf a(t)=-R\omega^2\cos(\omega t)\hat{\mathbf i}-R\omega^2\sin(\omega t)\hat{\mathbf j}=-\omega^2\mathbf r(t)\]

The position magnitude is always \(R\). The velocity magnitude is \(R|\omega|\), and the acceleration points toward the center with magnitude \(R\omega^2=v^2/R\). The dot product

\(\mathbf r\cdot\mathbf v=0\)

confirms that velocity is tangent to the circle.

At \(t=0\), the position is on the positive horizontal axis. For positive \(\omega\), velocity points upward and acceleration points inward to the left. These direction predictions are stronger checks than a visually plausible curve.

import numpy as np

radius_m, angular_speed, time_s = 2.0, 3.0, 0.5
phase = angular_speed * time_s
position = radius_m * np.array([np.cos(phase), np.sin(phase)])
velocity = radius_m * angular_speed * np.array([-np.sin(phase), np.cos(phase)])
acceleration = -(angular_speed**2) * position
print("Position (m):", position)
print("Velocity (m/s):", velocity)
print("Acceleration (m/s^2):", acceleration)
assert np.isclose(np.dot(position, velocity), 0.0, atol=1e-12)

The code uses the same phase for every component. Recomputing components at slightly different timestamps would create a state that never existed.

Geometry versus timing

Write a curve as \(\mathbf r(u)\) and let the parameter depend on time. The chain rule gives

\[\mathbf v=\frac{d\mathbf r}{du}\frac{du}{dt}\]

The first factor gives the curve's tangent with respect to the chosen parameter. The second controls how rapidly and in which direction the parameter changes. If \(du/dt=0\), the object is instantaneously at rest even if the curve has a well-defined tangent.

Differentiating again gives

\[\mathbf a=\frac{d^2\mathbf r}{du^2}\left(\frac{du}{dt}\right)^2+\frac{d\mathbf r}{du}\frac{d^2u}{dt^2}\]

The first term reflects the curve's bending, and the second reflects changing progress along the curve. Replacing time with another parameter without these chain-rule factors produces incorrect velocity and acceleration.

Tangential and normal acceleration

Where speed is nonzero, define the unit tangent \(\hat{\mathbf T}=\mathbf v/v\). Acceleration can be decomposed as

\[\mathbf a=\frac{dv}{dt}\hat{\mathbf T}+\frac{v^2}{\rho}\hat{\mathbf N}\]

The tangential part changes speed. The normal part changes direction and points toward the local center of curvature, with curvature radius \(\rho\). For uniform circular motion, the tangential part is zero and \(\rho=R\).

This decomposition becomes undefined where speed is zero because a direction of motion cannot be obtained by normalizing zero velocity. The Cartesian acceleration may still be perfectly well defined there.

Distance travelled and arc length

Displacement over an interval is \(\mathbf r(t_2)-\mathbf r(t_1)\). Distance travelled accumulates speed:

\[s=\int_{t_1}^{t_2}|\mathbf v(t)|\,dt\]

Distance is at least the displacement magnitude and can be larger when the path bends or reverses. After one complete circle, displacement is zero while distance is \(2\pi R\).

Numerical distance from samples can sum short segment lengths. That polygonal estimate usually underestimates a smooth curved path and should be checked by sampling more finely.

Differentiating sampled trajectories

If only position samples are available, centered differences estimate velocity at an interior sample:

\[\mathbf v(t_i)\approx\frac{\mathbf r(t_i+h)-\mathbf r(t_i-h)}{2h}\]

Apply the difference component by component, but use one shared time spacing. Uneven sampling needs formulas based on the actual timestamps. Differentiation amplifies measurement noise, especially for acceleration because it is a second derivative. A smoother-looking result produced by filtering is a model-dependent estimate and should not be presented as raw measurement.

times = np.linspace(0, 2, 201)
positions = radius_m * np.column_stack(
    [
        np.cos(angular_speed * times),
        np.sin(angular_speed * times),
    ]
)
velocities = np.gradient(positions, times, axis=0, edge_order=2)
print("Estimated velocity at 1 s (m/s):", velocities[100])

Domain, units, and continuity

Every parametric model needs a time domain. A formula valid from launch until impact must not be extended after impact without another motion phase. Piecewise curves need explicit boundary rules. Position should normally be continuous for ordinary motion, while idealized collisions can make velocity discontinuous.

Trigonometric function inputs should be dimensionless; in practice the phase \(\omega t\) is measured in radians. Position components use length units, velocity uses length per time, and acceleration uses length per time squared. A derivative routine that discards time units can return numerically plausible but dimensionally false values.

Checks for a trajectory function

Useful checks include:

Parametric motion unifies path and timing in one vector-valued function. Once that function is defined with a domain, frame, basis, and units, differentiation reveals how the position changes and integration reconstructs accumulated motion.

A worked Python notebook

Inspect the horizontal component of uniform circular motion with radius 2 metres and angular speed 3 radians per second. The question for this calculation is precise: how does horizontal position change when we change 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 vertical component is also needed to describe the full circular path. 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 2 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 time, expressed in s. The output is horizontal 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.

\[f(q) = 2 \cos{\left(3 q \right)}\]

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 0.5 s, the expression gives 0.141474 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*cos(3*q)", locals={"q": q})
reference_input = 0.5
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted horizontal 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 1 s, gives -1.97998 m. The upper endpoint gives 1.92034 m. Read these together with the assumptions: they describe different choices of 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, 1.0, 2.0])
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 time; the vertical axis shows horizontal 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, 2.0, 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("Time (s)")
ax.set_ylabel("Horizontal 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 horizontal position change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.5 s is -5.98497 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.

\[f'(q) = - 6 \sin{\left(3 q \right)}\]

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 = 0.51
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 0.51 s. The tangent estimate is 0.0816247 m, while the full expression gives 0.08157 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 = 2.0
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 time inside the stated interval, predict whether horizontal 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 vertical component is also needed to describe the full circular path. 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(2.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 2.0, 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

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