What it does
In a school textbook a thrown ball follows a perfect parabola and goes furthest at 45°. In real air it does not: drag slows it all the way, the path comes down more steeply than it went up, and the best angle is usually lower. This simulator works out the real path of a ball (or any roughly round object) with air drag that grows with the square of the speed, a steady wind along the direction of the throw, and a launch from any height.
It gives the range, the highest point and where it happens, the time of flight, the landing speed and angle, the path without air for comparison, the launch angle that gives the longest range with your drag and wind, and the terminal speed (the fastest the object could ever fall). It draws the path and the speed against time, and you can download the whole path as a CSV file for your own plots.
How to use it
- Pick an object to fill in its mass, size and drag coefficient, or type your own (the list switches to Custom).
- Type the launch speed and angle above level. A negative angle throws downwards (only from a height).
- Launch height is how far above the landing ground it leaves the hand or launcher.
- Mass and diameter set how much the air matters: a light, large object slows quickly; a heavy, small one hardly notices.
- Cd is the drag coefficient. Smooth spheres are roughly 0.1 to 0.5 depending on speed and surface. Set it to 0 to switch drag off and get the textbook parabola.
- Wind: positive is a tailwind (pushing the object along), negative a headwind.
- Air density (thinner air at altitude means less drag) and gravity (try 1.62 for the Moon).
How it works
The drag force is ½ ρ Cd A |v − w|² and points against the velocity relative to the air (v − w), where A is the frontal area from the diameter. Dividing by the mass gives the drag constant k = ρ Cd A / 2m: the slowing is k times the square of the air speed. The equations of motion have no closed-form solution, so the page steps them forward in time with the fourth-order Runge-Kutta method in 2,000 steps, and pins down the landing and the highest point exactly by bisection within the last step. The best angle is found by a golden-section search between 0.5° and 89.5°.
Worked example
The made-up small ball (145 g, 7.4 cm across, Cd 0.35) is thrown at 40 m/s at 35° from 1 m up in still air at sea level. The page shows a range of 94.85 m, a highest point of 21.25 m reached 51.75 m out after 1.91 s, and 4.102 s in the air. It lands at 24.8 m/s, 47.7° below level, much steeper than it was thrown. Without air the same throw would go 154.7 m and reach 27.84 m, so drag takes 38.7 % off the range. The best angle with this drag is 40.6°, giving 96.28 m. The terminal speed is 39.3 m/s, about the launch speed, which is why drag matters so much here. Type −5 in the wind box (a 5 m/s headwind) and the range drops to 84.41 m.
Tips
- If the terminal speed is far above the launch speed, drag is small and the no-drag answer is close. If it is near or below the launch speed, drag dominates.
- Real balls spin, and spin adds lift (the Magnus effect) that this model leaves out; golf and table-tennis balls are strongly affected.
- To compare with a video of a real throw, measure the path with a video or graph-digitising tool and overlay it on the downloaded CSV.
FAQ
Why is the best angle below 45°? Drag punishes long, high flights, so a flatter, faster path wins. A launch height also lowers the best angle, even without air.
Why does the ball come down steeper than it went up? It has lost horizontal speed to drag by the time it falls, so the descending path is more vertical.
Is the drag coefficient fixed? Here, yes. In reality Cd changes with speed (a sharp drop called the drag crisis for balls near 20 to 40 m/s, depending on roughness). Use a value suited to your speed range.
Is this accurate enough for sport or engineering? It is a sound physics model for study and planning, not a substitute for professional judgement or for measurements.
Where the numbers come from
- Newton's second law with quadratic drag F = ½ ρ Cd A v² and the fourth-order Runge-Kutta method: standard results in any classical mechanics or numerical methods text.
- Checked against an independent SciPy solve_ivp integration (relative tolerance 10⁻¹²) with event detection for landing and the highest point, and against the closed-form no-drag formulas.
- Air density 1.225 kg/m³ and standard gravity 9.80665 m/s² are rounded general values. The three example objects are made up.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.