Projectile motion is the path of an object thrown, launched, or fired once it is in free flight and only gravity acts on it. This calculator solves the ideal (no-air-resistance) equations for the four numbers people actually want — how far it goes, how high it climbs, how long it stays up, and how hard it lands — from just the launch speed, angle, and height. This guide explains the physics behind those numbers, why 45° is the magic angle on level ground, how a launch height changes that, and where the ideal model stops matching reality.
Horizontal and vertical motion are independent
The key insight that makes projectile motion solvable by hand is that the horizontal and vertical motions do not affect each other. Splitting the launch velocity into vₓ = v₀cosθ and v_y = v₀sinθ, gravity pulls only on the vertical part. Horizontally the projectile coasts at a constant vₓ; vertically it slows, stops at the apex, then speeds up on the way down — exactly like a ball thrown straight up. Because the two axes are independent, the time the projectile spends in the air is set entirely by the vertical motion, and the range is simply that time multiplied by the steady horizontal speed. That is why a bullet fired horizontally and a bullet dropped from the same height hit the ground at the same moment.
Why 45° maximizes range on level ground
On flat ground the range is R = v₀²·sin(2θ)/g. The factor sin(2θ) is largest when 2θ = 90°, i.e. θ = 45°, so 45° gives the maximum range for a given launch speed. There is a neat symmetry as well: complementary angles like 30° and 60°, or 40° and 50°, produce the same range — one trades a higher, slower arc for a flatter, faster one. The Compare Angles tab shows this directly. The trade-off is height versus distance: a steeper launch climbs higher and hangs longer but covers less ground, while a flatter launch reaches the target sooner along a lower path.
How a launch height shifts the best angle
The 45° rule only holds when the projectile lands at the same height it started. Launch from a cliff, a rooftop, or shoulder height and the optimal angle drops below 45°, given by θ_opt = atan(v₀/√(v₀² + 2gh₀)). The reason is that the extra height already buys hang time for free, so it pays to put more of the launch speed into horizontal motion. A shot put released from about 2 m, or a cannon on a 40 m cliff, both reach farthest a few degrees under 45°. The greater the height relative to the launch speed, the more the optimal angle falls — at very large heights it approaches a flat, horizontal launch.
Impact speed, gravity, and other worlds
Energy conservation gives a tidy result for landing speed: a projectile launched and landing at the same height returns at exactly its launch speed, just pointed downward at the mirror-image angle. Launch it from a height and it lands faster, because the drop adds kinetic energy. Gravity scales everything — range and maximum height are both inversely proportional to g, while time of flight is too. That is why the same throw carries about six times farther on the Moon (g ≈ 1.62 m/s²) and about 2.6 times farther on Mars (g ≈ 3.71 m/s²) than on Earth. Switching the gravity selector lets you explore these other-world trajectories instantly.
Ideal vs. real: the air-resistance caveat
This calculator uses the vacuum model: no air resistance, flat ground, and constant gravity. For heavy, slow, smooth objects over modest distances — a shot put, a thrown rock, a short cannon shot — it is very close to reality. But drag grows with speed and surface area, so for light or fast objects it overstates the result. A real baseball, golf ball, or arrow falls well short of the ideal range and its optimal launch angle is typically lower than 45° (often 30–40°) because a flatter, faster trajectory spends less time fighting drag. Spin adds lift or curve that the model ignores entirely. Treat these figures as the physics-textbook ideal and an upper bound on distance, not a ballistics-grade prediction.