Projectile Motion with Drag
Animated projectile with optional wind and drag. Summary text reports position, velocity, and energy.
Simulation not started.
Range Rider Mini-Game
Reinforce the launch range formula by firing rapid shots at moving pads. Drag, wind, and gravity shifts remix every run, so tweak angle and throttle on the fly to keep R = v² sin(2θ)/g inside the tolerance window before the timer drains.
Understanding Real-World Projectile Motion
Projectiles appear in sports, engineering, and natural phenomena. From a soccer ball curling through the air to a rock blasted from a volcano, the same physical principles dictate the arc. Traditional textbook treatments assume no air resistance, yet drag and wind often dominate the flight. This simulator keeps the familiar analytic case but adds a numerical integrator so you can watch how drag saps energy and wind skews the landing point. Variables are defined in SI units to maintain consistency: is mass in kilograms, the launch speed, the launch angle above the horizontal, gravitational acceleration, air density, a dimensionless drag coefficient, and the reference area. A steady wind adds a constant velocity to the air. These inputs combine to produce the motion shown on the canvas and summarized in the accessible text region.
Force Balance with Drag
The projectile experiences gravity and a drag force proportional to the square of relative velocity. The vector equation of motion is
where is the velocity relative to the air and its magnitude. The quadratic dependence arises from fluid dynamics for moderate Reynolds numbers. Horizontal and vertical components follow:
Integrating these coupled nonlinear equations analytically is difficult, so we employ a fourth-order Runge–Kutta (RK4) method for numerical stability. The timestep input allows experimenting with explicit scheme stability: large may cause energy drift, while very small steps increase computation time without appreciable benefit.
Numerical Scheme
The RK4 algorithm estimates the state = (, , , ) at successive times by sampling derivatives at intermediate points:
- Compute derivatives at the start of the interval.
- Estimate mid-interval values using half the timestep.
- Blend four derivative estimates to update position and velocity.
RK4’s error per step scales with , providing good accuracy for smooth forces. The code clamps between 0.001 s and 0.1 s to limit instability. If user inputs are invalid (NaN or negative where not physical), the script refuses to start and announces an error via the summary region. Each animation frame advances by one fixed step regardless of browser frame rate, ensuring deterministic trajectories.
Worked Example
Consider a 200 g steel ball ( kg) launched at m/s and °. With , m², kg/m³, and no wind, the drag coefficient yields an initial deceleration of roughly m/s² in the horizontal direction. Running the simulation shows the ball landing about 35 m away after 2.8 s, shorter than the 40 m predicted without drag. Switching on a 5 m/s tailwind extends the range to ~42 m, while a headwind cuts it to ~30 m. The CSV export confirms these numbers by listing time, position, velocity, and energies each timestep, enabling deeper analysis or plotting in external tools.
Comparison Table
The table contrasts three scenarios: baseline no-wind case, 5 m/s tailwind, and high drag (doubling ). Values are approximate.
| Case | Range (m) | Peak height (m) | Flight time (s) |
|---|---|---|---|
| Baseline | 35 | 10 | 2.8 |
| 5 m/s tailwind | 42 | 11 | 3.0 |
| High drag (Cd=0.94) | 26 | 7 | 2.5 |
How to Read the Animation
The projectile is drawn as a solid circle; its trail records past positions. The ground is the bottom canvas edge. A small green bar represents kinetic energy while a blue bar represents potential energy; the gray background shows total mechanical energy. When the projectile ascends, the blue bar grows as kinetic energy converts to potential; drag causes the combined bar length to shrink over time. Keyboard users can focus the canvas and press the spacebar to toggle play/pause. Tooltips appear when hovering over controls, and summary text below the canvas narrates the current time, position, velocity, and energies.
Limitations and Extensions
The model assumes a constant drag coefficient and air density, ignoring variations with speed and altitude. Spin-induced lift, turbulent crosswinds, and Coriolis effects are omitted. The integration neglects collisions with obstacles or the ground beyond first contact. Despite these simplifications, the simulator captures many key behaviors encountered in ballistics problems. Extensions could include stochastic wind gusts, user-defined drag laws, or 3‑D motion. Because all computation runs client‑side, performance may degrade on very slow devices for tiny timesteps; the checklist below suggests tests for stability at extreme settings.
References and Related Tools
Classical treatments of drag appear in standard texts such as J. B. Marion & S. T. Thornton, Classical Dynamics of Particles and Systems. The RK4 algorithm is described in most numerical analysis references. For related explorations, see the mass–spring–damper simulator, the Newton's second law calculator, or our orbital period simulation which also tracks energy drift.