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Projectile motion as simulation

Projectile motion is two-dimensional motion after an object is launched and before it hits another object or surface. The simplest model keeps only uniform gravitational acceleration near Earth's surface. It is valuable because its exact solution lets us check signs, event times, and numerical trajectory methods without an interactive simulation.

State the model before using the parabola

Choose a Cartesian frame with the positive vertical direction upward. During free flight, assume:

Under these assumptions,

\[\mathbf a=(0,-g)\]

The object's mass does not appear because gravitational force \(m\mathbf g\) divided by mass gives the same acceleration for every projectile in this ideal model. With drag, shape and mass can matter.

Initial state and component equations

At launch time \(t=0\), let the position be \((x_0,y_0)\). If the initial speed is \(v_0\) and the launch angle measured above the positive horizontal axis is \(\theta_0\), then

\[v_{0x}=v_0\cos\theta_0,\qquad v_{0y}=v_0\sin\theta_0\]

Use radians in programming-language trigonometric functions unless the API explicitly says otherwise. The sign and quadrant of the components follow the angle convention; an angle alone is ambiguous if its reference axis and positive direction are omitted.

Integrating the constant acceleration gives

\[v_x(t)=v_{0x},\qquad v_y(t)=v_{0y}-gt\]

and

\[x(t)=x_0+v_{0x}t,\qquad y(t)=y_0+v_{0y}t-\frac12gt^2\]

These two functions define the trajectory during free flight, not the motion after impact. Their valid time domain ends at the first relevant collision. Here “simulation” means evaluating or numerically approximating this declared model; there is no interactive simulation on the page.

import numpy as np


def projectile_state(initial, speed, angle, time, gravity=9.81):
    if time < 0 or gravity <= 0:
        raise ValueError("Use nonnegative time and positive gravity.")
    vx, vy = speed * np.cos(angle), speed * np.sin(angle)
    position = np.asarray(initial) + [vx * time, vy * time - 0.5 * gravity * time**2]
    velocity = np.array([vx, vy - gravity * time])
    return position, velocity


print(projectile_state([0.0, 0.0], 10.0, np.pi / 4, 1.0))

Why the ideal path is a parabola

If \(v_{0x}\ne0\), solve the horizontal equation for time and substitute into the vertical equation:

\[y(x)=y_0+\frac{v_{0y}}{v_{0x}}(x-x_0)-\frac{g}{2v_{0x}^2}(x-x_0)^2\]

The negative quadratic coefficient makes a downward-opening parabola. This path result depends on constant downward acceleration and zero horizontal acceleration. Air drag makes the trajectory generally nonparabolic, and a vertical launch with \(v_{0x}=0\) cannot use this eliminated-time formula even though its time equations remain valid.

Highest point

At the highest point of an upward launch, vertical velocity is zero:

\[t_{\mathrm{top}}=\frac{v_{0y}}{g},\qquad v_{0y}>0\]

Substitution gives the maximum height

\[y_{\max}=y_0+\frac{v_{0y}^2}{2g}\]

At this instant only the vertical velocity is zero. Horizontal velocity remains \(v_{0x}\), so the projectile is not generally at rest. If \(v_{0y}\le0\), the maximum over the post-launch free-flight interval may occur immediately at launch, depending on the landing surface.

Landing time on a horizontal level

Suppose the projectile first intersects a horizontal surface at \(y=y_f\). Solve

\[y_f=y_0+v_{0y}t-\frac12gt^2\]

The quadratic roots are

\[t=\frac{v_{0y}\pm\sqrt{v_{0y}^2+2g(y_0-y_f)}}{g}\]

Only real roots within the modeled time domain are candidates. A negative root is usually a mathematical time before launch. When the launch point lies on the landing height, one root is \(t=0\); landing normally means the later positive root. A discriminant below zero means the ideal trajectory never reaches that height.

def landing_time(initial_height, initial_vertical_velocity, level=0, gravity=9.81):
    discriminant = initial_vertical_velocity**2 + 2 * gravity * (initial_height - level)
    if discriminant < 0:
        return None
    roots = [
        (initial_vertical_velocity - np.sqrt(discriminant)) / gravity,
        (initial_vertical_velocity + np.sqrt(discriminant)) / gravity,
    ]
    future = [root for root in roots if root > 1e-12]
    return min(future) if future else None


time = landing_time(0.0, 10 * np.sin(np.pi / 4))
print(f"Landing time: {time:.3f} s")

The “first landing” logic is a physical event rule layered on the quadratic equation. If terrain varies with horizontal position, solve the intersection between the trajectory and terrain function instead.

Range and its limited shortcut

Horizontal range to the selected landing time is

\[R=x(t_{\mathrm{land}})-x_0=v_{0x}t_{\mathrm{land}}\]

When launch and landing heights are equal, the launch is upward, and the ideal assumptions hold, this simplifies to

\[R=\frac{v_0^2\sin(2\theta_0)}{g}\]

This familiar formula is not valid for different launch and landing heights, nonlevel terrain, drag, or a target struck before ground level. Under its assumptions and for a fixed speed, complementary angles have the same range, and the largest range occurs at \(45^\circ\). These are useful checks, not universal statements about real projectiles.

Worked numerical case

Launch from

\((x_0,y_0)=(0,5)\,\mathrm m\)

with velocity components

\((v_{0x},v_{0y})=(8,6)\,\mathrm{m/s}\)

and take \(g=9.8\,\mathrm{m/s^2}\). For ground at \(y_f=0\), the positive root is

\[t_{\mathrm{land}}=\frac{6+\sqrt{6^2+2(9.8)(5)}}{9.8}\approx1.79\,\mathrm s\]

The range is approximately \(8(1.79)=14.3\,\mathrm m\). The top occurs at about

\(6/9.8=0.612\,\mathrm s\)

and reaches about \(5+6^2/[2(9.8)]=6.84\,\mathrm m\).

Energy provides an independent check during free flight:

\[E=\frac12m(v_x^2+v_y^2)+mgy\]

This energy should remain constant before impact. The arbitrary potential zero changes the reported potential energy but not energy differences or the trajectory.

Numerical stepping and exact checks

A numerical solver may be required after adding velocity-dependent drag or wind. For the ideal constant-acceleration case, however, the analytic equations are the reference. Compare numerical positions and velocities at identical physical times, repeat with a smaller step, and verify that errors decrease as expected for the method.

Event detection matters. If a step changes vertical position from above terrain to below it, reporting the below-ground endpoint gives a step-dependent range. Locate the crossing within the step by interpolation or root finding and stop at the event time. With curved motion or acceleration, linear interpolation is approximate and should be refined.

What changes when drag is included

A common drag model adds a force opposite velocity. Linear drag is proportional to speed, while quadratic drag has the vector form

\[\mathbf F_D=-c|\mathbf v|\mathbf v\]

The coefficient depends on fluid density, shape, area, and drag coefficient. Horizontal and vertical motion are now coupled through speed \(|\mathbf v|\). The horizontal velocity is no longer constant, the path is not generally a parabola, and the same- height range formula does not apply. A numerical solution needs a stated drag law and parameters rather than an unexplained visual adjustment.

Checks that catch common errors

The ideal projectile model is simple because vector components reduce it to two linked descriptions sharing one time. Its exact answers make it a powerful test case, provided the page does not mistake a limited parabola for every real flight.

A worked Python notebook

Launch a projectile at 10 metres per second and 45 degrees from level ground. The question for this calculation is precise: how does height change when we change time since launch, 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.

This interval stops before landing; air resistance and changes in gravity are omitted. 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 1.4 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 time since launch, expressed in s. The output is height, 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.

\[f(q) = - 4.905 q^{2} + 5 \sqrt{2} q\]

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 0.7 s, the expression gives 2.5463 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("10*sin(pi/4)*q-9.81*q**2/2", locals={"q": q})
reference_input = 0.7
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted height:", 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 0 m. The midpoint, at 0.7 s, gives 2.5463 m. The upper endpoint gives 0.285695 m. Read these together with the assumptions: they describe different choices of time since launch 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, 0.7, 1.4])
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 since launch; the vertical axis shows height. 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, 1.4, 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 since launch (s)")
ax.set_ylabel("Height (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 height change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.7 s is 0.204068 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.

\[f'(q) = - 9.81 q + 5 \sqrt{2}\]

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 = 0.707
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 0.707 s. The tangent estimate is 2.54773 m, while the full expression gives 2.54749 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 = 1.4
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 since launch inside the stated interval, predict whether height 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: This interval stops before landing; air resistance and changes in gravity are omitted. 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(1.4) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 1.4, 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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