M003 · Ballistics / inverse design

Projectile Flight Designer

A launch platform and movable ground target connect side elevation, crosswind plan view, velocity components and energy accounting. Compare vacuum, linear drag and quadratic drag; steer around a crosswind or search for the best downrange elevation at fixed launch speed.

Elapsed time $0\,\mathrm{s}$

Velocity: downrange, vertical, crossrange $\begin{bmatrix}0\\0\\0\end{bmatrix}\,\mathrm{m\,s^{-1}}$

Acceleration: downrange, vertical, crossrange $\begin{bmatrix}0\\0\\0\end{bmatrix}\,\mathrm{m\,s^{-2}}$

Landing time $0\,\mathrm{s}$

Downrange landing $0\,\mathrm{m}$

Crossrange landing $0\,\mathrm{m}$

Ground target miss $0\,\mathrm{m}$

Trajectory curvature $0\,\mathrm{m^{-1}}$

Mechanical energy $0\,\mathrm{J}$

Work done by air $0\,\mathrm{J}$

Energy balance defect $0\,\mathrm{J}$

Landing change with half step $0\,\mathrm{mm}$

Design resultAdjust launch or drag target

LAUNCH / COMPARE / DESIGN

Make the target, audit the flight

Vacuum referenceSame launch, gravity and mass; air forces removed.Velocity and acceleration arrowsBlue is velocity; red is total acceleration.Launch speed

v_0

25.00\,\mathrm{m\,s^{-1}}

Launch elevation

\theta

42.00\,{}^\circ

Launch bearing

\phi

0.00\,{}^\circ

Platform height

h_0

2.00\,\mathrm{m}

Local gravity

g

9.81\,\mathrm{m\,s^{-2}}

Particle mass

m

0.15\,\mathrm{kg}

Air model: vacuum / linear / quadratic

\mathcal{D}

0.00\,

Drag coefficient multiplier

\lambda_D

1.00\,

Downrange wind

w_x

0.00\,\mathrm{m\,s^{-1}}

Crosswind

w_z

0.00\,\mathrm{m\,s^{-1}}

Target downrange

x_T

60.00\,\mathrm{m}

Target crossrange

z_T

0.00\,\mathrm{m}

Maximum RK4 step

\Delta t_{\max}

20.00\,\mathrm{ms}

Inspect flight

t/t_{\mathrm{impact}}

0.00\,\%

Drag the purple launch handle for speed and elevation. Drag the gold target in the plan view for both ground coordinates. Keyboard sliders provide the same controls. Wind acts through air-relative velocity; it has no effect in vacuum.

A three-dimensional point particle in uniform gravity and steady horizontal wind. Linear and quadratic drag are separate phenomenological models with constant coefficients, not a Reynolds-number transition. No lift, spin, buoyancy, density gradient, terrain, bounce or Earth curvature. RK4 uses a drag-aware step limit and solves the landing event; halving the step is a convergence check, not a rigorous error bound. Target tolerance is 2 m. Search is bounded to the shown controls and may report a miss.

Physics tutorial

From a trajectory to a launch design

BackgroundProjectile motion is an initial-value problem when the launch is known, and an inverse design problem when a target is specified. Here the same three-dimensional trajectory drives side elevation, ground-plan drift, velocity components and the energy ledger. The target lies on the flat ground; the first descending ground crossing ends the experiment.

Why it mattersOpenStax University Physics, Volume 1, sections 4.3 and 6.4 establish the vacuum benchmark and quadratic-drag force law. The workbench adds a separate linear-drag model, air-relative velocity and a bounded numerical design search so every change has a measurable consequence.

Start with the essentials

Focus question: Can a launch chosen in vacuum still hit the target after a crosswind and drag are added?
One-sentence intuition: Gravity acts vertically, but drag depends on the full air-relative velocity $\boldsymbol u=\boldsymbol v-\boldsymbol w$ . A side view can conceal a substantial crossrange miss; the plan view closes that measurement loop.

Core mathematical model

Three-dimensional equation of motion

\dot{\boldsymbol r}=\boldsymbol v,\qquad m\dot{\boldsymbol v}=-mg\boldsymbol e_y+\boldsymbol F_D

The coordinates are downrange, vertical and crossrange. The launch elevation and bearing set all three initial velocity components. RK4 advances the point particle, with the maximum step reduced when drag is strong.

Choose one air model

\boldsymbol F_D=\begin{cases}\boldsymbol 0&\text{vacuum},\\-b\boldsymbol u&\text{linear},\\-k\lVert\boldsymbol u\rVert\boldsymbol u&\text{quadratic},\end{cases}\qquad\boldsymbol u=\boldsymbol v-\boldsymbol w

The multiplier scales a reference linear coefficient of 0.08 kg per second or a quadratic coefficient of 0.003 kg per metre. These are controlled phenomenological coefficients; the slider does not compute Reynolds number or a material-specific drag coefficient.

Vacuum time and range benchmark

t_{\mathrm{impact}}=\frac{v_0\sin\theta+\sqrt{v_0^2\sin^2\theta+2gh_0}}{g},\qquad x_{\mathrm{impact}}=v_0\cos\theta\cos\phi\,t_{\mathrm{impact}}

For a level launch with zero bearing and no air forces, the maximum range occurs at 45 degrees. Launch height and air forces change that optimum; mass cancels only in vacuum. Wind settings must leave the vacuum trajectory unchanged.

Energy and curvature diagnostics

E=\tfrac12m\lVert\boldsymbol v\rVert^2+mgy,\quad W_{\mathrm{air}}=\int\boldsymbol F_D\cdot\boldsymbol v\,dt,\quad\kappa=\frac{\lVert\boldsymbol v\times\boldsymbol a\rVert}{\lVert\boldsymbol v\rVert^3}

Mechanical energy minus its initial value equals work done by air in the ground frame. A wind can add mechanical energy, so a positive air-work value is not automatically a numerical failure. Curvature describes turning per unit path length, not speed alone.

Common difficulties

Wind is not an extra constant force

Typical misconceptionA wind simply adds a fixed sideways acceleration, even in vacuum.

Better mental modelAir forces respond to the velocity relative to the wind and disappear in vacuum. Changing mass at fixed drag coefficients changes acceleration because force is divided by mass.

A numerical search is not a promise

Typical misconceptionFind target launch guarantees that every target can be reached.

Better mental modelThe search holds speed and environment fixed, and stays within the elevation and bearing limits. A closest shot outside the 2 m target tolerance is reported as a miss; an unreachable target is not silently moved.

Step refinement is evidence, not a bound

Typical misconceptionThe half-step landing difference is the exact error in the trajectory.

Better mental modelIt measures sensitivity to a finer step. The vacuum analytic benchmark and linear-drag tests provide independent checks; neither a small difference nor a passing energy ledger proves an omitted physical effect is negligible.

Run the experiment

01
Recover the vacuum benchmark
Choose Vacuum range, set platform height and bearing to zero, then maximize downrange. Vary mass and both winds.
What to observe: The optimum approaches 45 degrees; mass and wind do not change the vacuum solution. The event-resolved landing time agrees with the analytic reference.
02
Make the air visible
Compare Linear drag and Crosswind challenge. Pause near the apex and inspect the plan view, velocity components and air work.
What to observe: The two air models produce different trajectories. A crosswind builds a crossrange displacement that a side view cannot reveal.
03
Close the inverse-design loop
Drag the gold target in the plan view, then find a target launch. Move it far beyond the available range and repeat.
What to observe: The search adjusts elevation and bearing at fixed speed. Reachable targets are hit within tolerance; out-of-range designs remain visibly missed.
04
Audit height and convergence
Choose Cliff launch, maximize downrange, and compare maximum step sizes while watching the half-step landing difference.
What to observe: The optimum need not be 45 degrees. Strong drag triggers smaller stability steps; the landing event is found inside the final integration step.