Projectile Motion Lab
Explore 2D point-mass motion from an initial height to flat ground. A drag-free analytic solution supplies a baseline. A fixed-step RK4 solver applies an explicitly chosen constant effective quadratic-drag coefficient, then reruns with half and quarter time steps so you can inspect the observed numerical change. Scrub position through time and export the sampled trajectory. This is an educational model, not a certified launch or safety calculation.
Key features
- Vacuum analytic flight time, range and peak from the same initial conditions
- Quadratic-drag numerical path using fixed-step fourth-order Runge–Kutta integration
- Time-step, half-step and quarter-step impact comparison plus vacuum numerical error
- Interactive trajectory, height-vs-time and distance-vs-time graphs with a scrubber
- Time, position and velocity sample table; local CSV, report JSON and SVG downloads
- Explicit SI units, model assumptions and bounded input/horizon errors
How to use
- Start with the example or enter SI speed, launch angle, initial height and gravity.
- Choose no drag or effective quadratic drag, set its per-meter coefficient and a numerical time step.
- Run the model to compare the analytic vacuum baseline with the numerical outcome and impact metrics.
- Inspect the three time-step results and scrub or play through position and velocity samples.
- Download all computed samples as CSV, a JSON report, or the trajectory drawing as SVG.
Use cases
- See how a chosen drag assumption changes a classroom projectile trajectory
- Study the effect of halving a numerical integration interval on impact estimates
- Inspect velocity and height over time before documenting a simple model
Frequently asked questions
What does the drag coefficient mean?
The effective beta has units m⁻¹ and gives acceleration −beta·|v|·v, with v in m/s. Under constant air density, drag coefficient, reference area and mass, beta equals rho·Cd·A/(2m). You enter beta directly; the app does not infer a physical object's properties.
What assumptions are made?
The model is a two-dimensional point mass over flat ground y=0 with constant gravity, constant effective drag, no wind, thrust, lift, spin, terrain or varying air density. The no-drag path is an analytic reference for those conditions only.
Is this an exact solution with air resistance?
No. Drag motion is approximated by fixed-step fourth-order Runge–Kutta. Impact is interpolated linearly between the last two samples. The table shows dt, dt/2 and dt/4 estimates; a decreasing difference supports but does not prove convergence or physical accuracy.
Why can a run be rejected?
Inputs outside the documented SI ranges, a time step too coarse for the selected drag, non-finite values, or a flight longer than the 180-second horizon are rejected. Reduce the step or model range; no result is silently truncated.
What is the difference between the blue and dashed paths?
Blue is the selected numerical model. The gray dashed path is the drag-free analytic trajectory from the same speed, angle, height and gravity. When drag is off the paths overlap except near interpolated impact.
Can I use these numbers for equipment design or safety?
No. The model omits wind, rotation, lift, changing atmospheric density and many object-specific effects. Numerical agreement between time steps is not a validation of the assumptions or a professional design assurance.
Privacy
All calculations are performed in the current browser tab. CSV, JSON and SVG are generated only when you download them. No project data is uploaded or persistently stored.
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