Projectile motion simulator comparing three physical regimes: vacuum (no air), Stokes drag (linear in velocity, F=bv, laminar flow), and Newton drag (quadratic, F=½ρCdAv², turbulent regime). In vacuum the optimal angle for maximum range is exactly 45°. With drag the optimum drops to 30–38° depending on drag magnitude. The simulator numerically integrates the equations of motion with RK4 and shows both the multi-trajectory angle sweep and the live comparison between all three regimes.
This simulation plots the flight of a projectile launched from the ground and compares an idealised vacuum trajectory against one slowed by air resistance. In vacuum the path is a perfect parabola governed by gravity alone (g = 9.81 m/s²). With air, a Newton drag force, F = ½ρCdAv², acts opposite to velocity. The equations of motion are integrated numerically with a small time step (dt = 0.005 s), so the curved, asymmetric real-world path emerges directly.
You choose a projectile — football, arrow or cannonball, each with its own mass, radius and frontal area — and set the launch angle (5–85°), muzzle speed (10–100 m/s) and drag coefficient Cd (0–1). The telemetry panel reports range with and without drag, peak height, flight time and the optimal angle for maximum range, which drag pushes below 45°. The same physics governs artillery, ballistic sport and long-range marksmanship.
What does this simulation actually show?
It draws the trajectory of a launched projectile in two regimes side by side: a dashed vacuum parabola and a solid curve that includes air resistance. As you change the angle, speed, projectile and drag coefficient, both paths and the live telemetry (range, height, flight time and optimal angle) update instantly so you can see how air resistance reshapes the flight.
Why is the optimal launch angle less than 45 degrees?
In a vacuum, 45° gives the longest range because horizontal and vertical motion are symmetric. Drag grows with the square of speed, so it bites hardest early in the flight when the projectile is fastest. Launching a little flatter — typically 30–40° depending on drag — keeps more horizontal speed and lengthens the carry, which is why the simulator's optimal angle drops below 45°.
What is the difference between Stokes and Newton drag?
Stokes drag is linear in velocity (F = bv) and applies to slow, laminar flow such as tiny particles in fluid. Newton drag is quadratic (F = ½ρCdAv²) and dominates for fast, turbulent flow — balls, arrows and cannonballs in air. This simulation uses the Newton quadratic model, which is the realistic regime for everyday projectiles.
The projectile buttons pick a football, arrow or cannonball, each setting a different mass, radius, frontal area and default drag coefficient. The launch-angle slider sets the firing elevation (5–85°), the speed slider sets the initial muzzle velocity (10–100 m/s), and the Cd slider scales the drag coefficient from 0 (frictionless) up to 1. The Angle Sweep button overlays trajectories at 10° intervals.
Each step solves Newton's second law with two forces: gravity, giving a constant downward acceleration g = 9.81 m/s², and drag, giving an acceleration of magnitude (½ρCdA/m)·v² directed against the velocity vector. With air density ρ = 1.225 kg/m³, the simulator advances velocity and position by a small fixed time step until the projectile returns to the ground.
The model captures the dominant physics — gravity and quadratic air drag — and uses realistic constants for air density and each projectile's mass and area. It is a good qualitative and semi-quantitative guide. It omits secondary effects such as wind, spin and the Magnus force, lift, projectile tumbling, and the variation of air density and gravity with altitude.
Drag deceleration scales as drag force divided by mass. A 5 kg cannonball has huge inertia relative to its frontal area, so air resistance barely slows it and its path stays close to the vacuum parabola. A 25 g arrow has very little mass, so the same drag force decelerates it strongly. Heavy, dense, streamlined projectiles keep their speed and range much better.
It overlays a fan of trajectories computed at fixed angle intervals from 10° to 80°, all using your current speed, projectile and drag settings. This lets you see at a glance which firing angle produces the greatest range and how the envelope of reachable points changes, making the optimal-angle result visual rather than just a number.
Without air, ascent and descent mirror each other perfectly. Drag continually removes energy, so the projectile climbs faster than it falls and its horizontal speed keeps dropping. The result is a lopsided curve with a steeper, shorter descent than ascent — the projectile drops more sharply near the end of its flight than it rose at the start.
The same drag-and-gravity model underlies artillery and mortar fire-control tables, the carry of a golf ball or long throw in sport, archery and long-range rifle ballistics, and the design of catapults and trebuchets. Engineers and athletes all exploit the fact that, in air, a slightly flatter launch than 45° maximises distance.