Projectile Motion Simulator
Adjust the launch angle, the power, and how heavy the ball is, and watch what the air does to the throw. It's the same launch-angle intuition behind the paper airplane flight lab — though read the honest caveat below before assuming it explains the airplane itself.
- Range (with air)
- 75.8 m
- Range (no air)
- 124.9 m
- Lost to air
- 39.3%
- Flight time
- 4.34 s
- Peak height
- 23.3 m
- Launch speed
- 35 m/s
Both paths come from the same step-by-step solver. Running it with the air switched off should reproduce the textbook range of 124.9 m exactly — it lands within <0.001%, which is what makes the air-resistance path worth believing, since no formula exists to check that one against.
Assumes a smooth sphere, still air at sea level, and no spin. Spin matters enormously in real sport — backspin is most of why a golf ball carries — and isn't modelled here.
0.15 kg · ⌀ 7 cm
The math behind it
If gravity were the only force acting on a thrown ball, working out where it lands would take one line. That's the formula physics class starts with, and it describes a throw in a vacuum:
range = (speed² × sin(2 × angle)) ÷ gravity
That formula is also why 45° gives the longest range in a vacuum — sin(2 × angle) peaks at exactly 90°, which happens when angle = 45°. Launch flatter or steeper than that and you trade horizontal distance for either less hang time or more.
What the air does to it
Air is not a vacuum. A ball pushing through it feels a drag force that grows with the square of its speed — double the speed and the air pushes back four times as hard. Turn the launch power up and watch the gap between the two paths open: the dashed line is the textbook answer, the solid line is what the ball actually does.
The mass slider is the interesting one. A heavy ball and a light ball of the same size meet exactly the same air force, but the heavy one has far more inertia to resist it, so it flies closer to the vacuum path. That ratio — how much ball there is compared with how much air it has to shove aside — is why a golf ball carries and a ping-pong ball stops dead. It's also why the best launch angle drops below 45° once air is involved: a flatter throw spends less time being slowed down.
Adding drag means giving up the tidy formula. It couples the horizontal and vertical motion together — the sideways drag depends on how fast the ball is falling, and vice versa — and there's no clean closed-form solution to that. So the simulator solves the motion step by step instead, with fourth-order Runge-Kutta, advancing position and velocity through small slices of time.
That raises a fair question: if there's no formula to compare against, how do you know the answer is right? By running the same solver with the air switched off, where a formula does exist. It reproduces the textbook range to under a thousandth of a percent, which is printed under the diagram. The solver is sound; it's only the physics that changed.
One honest caveat: this models a ball, not a paper airplane. A paper airplane's flight is shaped by aerodynamic lift from its wings, a force this model doesn't include at all, which is exactly why a well-designed one glides much farther and flatter than any ballistic path would predict. Treat this as the launch-angle intuition underneath the paper airplane challenge, not a model of the airplane itself.
More on forces as a concept →