Projectile motion is the textbook ballistic problem: an object launched at velocity v and angle θ, subject only to gravity, follows a parabolic arc. The Toolenza calculator returns range = v² · sin(2θ) / g, peak height = (v · sin θ)² / (2g), time of flight = 2v · sin θ / g, and landing speed.

The non-obvious result

Range is maximised at exactly 45° launch angle — but only in a vacuum. With air drag, the optimum drops to 30–40° for a baseball, ~35° for a soccer ball. The textbook 45° answer is correct only when you neglect air resistance, which this calculator does.

Where it applies

• Physics homework — the canonical intro-mechanics problem.
• Game design — projectile arcs in games like Angry Birds use exactly this formula before drag is layered in.
• Quick estimates — back-of-envelope answer for "how far can I throw / launch / shoot something".

What it ignores

Air resistance, the Magnus effect (spin-induced lift), wind, and altitude-varying gravity. For real-world ballistics — artillery, rocketry, even competitive throwing — you need a numerical solver with a drag model. This calculator is the first approximation, not the production answer.