Arrow Trajectory Simulator

A physics-based GUI simulation (Java + JavaFX) for calculating and optimizing arrow trajectories under realistic conditions: gravity, quadratic aerodynamic drag, and wind.

What it does

• Simulates an arrow's flight using 4th-order Runge-Kutta (RK4) numerical integration of the equations of motion (not the simplified no-air-resistance formula most projectile demos use).
• Visualizes the trajectory on an animated canvas, with a target you can place at any distance/height.
• Optimizes:
  • Max range: golden-section search finds the launch angle that maximizes range for a given speed (this is not 45° once drag is included — the simulator finds the real optimum).
  • Target solving: given a target's distance and height, finds every launch angle at the given speed that lands on it (typically a "flat" low-arc shot and a "lobbed" high-arc shot, when both exist).

Project structure

src/main/java/com/arrowtrajectory/
├── Main.java                       Plain launcher (see note below)
├── physics/
│   ├── Vector2D.java                Small 2D vector helper
│   ├── ArrowProperties.java         Mass, diameter, drag coefficient
│   ├── EnvironmentProperties.java   Gravity, air density, wind speed
│   ├── TrajectoryPoint.java         One sampled (t, x, y, vx, vy) state
│   ├── SimulationResult.java        Full trajectory + summary stats
│   ├── TrajectorySimulator.java     RK4 integrator (the physics core)
│   └── OptimizationEngine.java      Max-range & target-solving search
└── gui/
    ├── MainApp.java                 JavaFX Application, wires everything up
    ├── ControlPanel.java            Left-hand input sliders/fields
    ├── TrajectoryCanvas.java        Draws + animates the trajectory
    └── ResultsPanel.java            Bottom stats strip

src/test/java/.../physics/          JUnit 5 tests (validated against the
                                     analytic no-drag projectile formula)
src/main/resources/style.css        Dark theme stylesheet

Why Main.java exists separately from MainApp: running a JavaFX Application subclass as the main class of an executable "fat" jar (built via shade/assembly) without the JavaFX SDK on the module path can throw Unable to open DISPLAY-adjacent runtime errors on some setups. A plain launcher class that just calls MainApp.main(args) avoids that class of problem — this is a well-known JavaFX packaging workaround, not a bug.

The physics

Two forces act on the arrow after release:

1. Gravity — constant, downward: a_gravity = -g

2. Aerodynamic drag — quadratic in relative airspeed (i.e. it accounts for wind):

   F_drag = 0.5 * ρ * Cd * A * |v_rel| * v_rel
   

   where ρ is air density, Cd the drag coefficient, A the arrow's cross-sectional area, and v_rel = v_arrow - v_wind.

Because drag depends nonlinearly on velocity, the simulator integrates the equations of motion with RK4 at a 1ms step (rather than simple Euler integration), which stays accurate over the arrow's full flight without needing an unreasonably tiny step size.

The max-range search works because range-vs-angle is unimodal (it rises, peaks, then falls) even with drag — so golden-section search reliably converges on the true optimum in a few dozen evaluations.

The target solver doesn't assume the height-vs-angle function is unimodal; instead it samples angles finely, looks for sign changes in height_at_target_distance - target_height, and bisects each bracket it finds. That's why it can report both a low-arc and high-arc solution for the same target.

Building & running

Requires JDK 17+ and Maven. JavaFX dependencies are pulled from Maven Central automatically — no manual SDK download needed.

# Run directly (recommended during development)
mvn clean javafx:run

# Or build a runnable fat jar and run it
mvn clean package
java -jar target/arrow-trajectory-simulator-1.0.0.jar

# Run the test suite
mvn test

Using the app

1. Adjust Initial Speed, Launch Angle, and Launch Height on the left, plus the arrow's Mass, Shaft Diameter, and Drag Coefficient, and environmental Wind Speed / Air Density.
2. Click Simulate & Animate to fly the arrow and see the range, max height, flight time, and impact speed.
3. Click Find Max-Range Angle to have the optimizer pick the best angle for the current speed/arrow/environment.
4. Check Aim at a target, enter a distance and height, and click Solve Angle(s) For Target to find the angle(s) that hit it.

Notes on realism

Default arrow values (26g mass, 6mm shaft, Cd ≈ 1.2) approximate a typical fletched target/hunting arrow. Air density and gravity default to sea-level standard conditions but are both adjustable. This is an educational/portfolio simulation, not a certified ballistics tool.