Thrown, and falling

Projectile motion is the motion of any object that is launched and then moves under gravity alone — a thrown ball, a kicked football, an arrow, or a jet of water. Once it leaves the hand or launcher, the only force on it, ignoring air, is gravity pulling straight down. The object coasts sideways at a steady speed while it rises and falls, and the combination of these two motions traces a smooth curve called a parabola. Splitting the motion into a steady horizontal part and a falling vertical part is the key to understanding it.

Because two different kinds of motion combine. Horizontally there is no force (ignoring air), so the object moves at a constant velocity, covering equal distances in equal times. Vertically, gravity accelerates it downward, so its drop increases with the square of the time. When one quantity grows steadily while another grows as the square of the time, the resulting curve is a parabola — the same shape as a "y depends on x squared" graph in mathematics. Galileo was the first to prove, in the early 1600s, that projectiles follow parabolas.

Because their vertical motions are identical and independent of any sideways motion. Both fall the same height under the same gravity, accelerating downward at g, so both take exactly the same time to reach the ground. The thrown ball also travels sideways, but there is no horizontal force to change its fall, and moving sideways does nothing to speed up or slow the drop. Since the time to land is set entirely by the vertical fall, and both fall identically, they land together. This independence of horizontal and vertical motion is the heart of projectile analysis.

It means a projectile's horizontal and vertical motions proceed completely separately, neither affecting the other. The horizontal motion is steady, an example of inertia with no force; the vertical motion is free fall under gravity. You can work out each one on its own — how far it goes sideways, and how long it takes to fall — and then combine them to get the full curved path. This idea, due to Galileo, is what makes projectile motion solvable: instead of one complicated curve, you have two simple, familiar motions side by side.

On level ground, ignoring air, a launch angle of 45° gives the maximum range. The range is given by R = speed² × sin(2 × angle) ÷ g, and sin reaches its largest value of 1 when twice the angle is 90°, that is, at 45°. Below 45° the projectile is flatter and lands too soon; above it, it climbs high but does not travel as far. A neat consequence is that complementary angles — any pair adding to 90°, such as 30° and 60° — give exactly the same range, though with different heights and flight times.

In the ideal case, ignoring air, no — mass makes no difference to the path. Just as all objects fall at the same rate in a vacuum, all projectiles launched at the same speed and angle follow the same parabola, whatever their mass. A heavier object feels more gravity but also has more inertia, and the two cancel. In real air, though, mass and shape do matter: light or oddly shaped objects are slowed more by drag, so a feather and a cannonball launched alike would part ways. The clean parabola is the no-air idealisation.

At the peak, the projectile's vertical velocity is momentarily zero — gravity has just cancelled the upward motion — but its horizontal velocity carries on unchanged, so the object is still moving sideways, not at rest. Its acceleration is still the full g downward, which is exactly what is about to start it falling. This is a good example of how velocity and acceleration are independent: the velocity can be zero in one direction while the acceleration is not zero at all. The top of the arc is the slowest point of the flight, but never a stationary one.

It is a classic demo showing the independence of motion. A launcher is aimed directly at a target monkey, which is released to fall the instant the launcher fires. Remarkably, the dart always hits the monkey, whatever the firing speed. The reason is that, once released, both the dart and the monkey fall at the same rate, g. In the time the dart travels, it drops below its straight-line aim by exactly the same amount the monkey drops from the branch, so the two meet. Aiming straight at the monkey is always correct, because gravity treats them identically.

Air resistance is a force opposing the motion, so it robs a projectile of speed throughout its flight. The result is a shorter range than the ideal parabola predicts, and a path that is no longer symmetric: the descent is steeper and slower than the ascent. The faster or lighter the object, the bigger the effect — which is why a thrown feather behaves nothing like a thrown stone. Because drag bites hardest at high speed, the best angle for maximum range in real air is usually a little below 45°, not exactly 45°.

It is simply those two motions happening at once. In the horizontal direction there is no force, so the projectile coasts at a constant velocity — exactly inertia, the same steady motion a disc shows on a frictionless table. In the vertical direction, gravity accelerates it downward at g — exactly free fall, the same motion a dropped object undergoes. Because the two directions are independent, you can treat one as pure inertia and the other as pure free fall, then combine them. The parabola is what you get when steady sideways coasting meets an accelerating drop.

In a sense, yes — that is exactly what an orbit is. On flat ground a projectile always lands, because the Earth stays level beneath it. But the Earth is curved, and if you could launch a projectile fast enough, around 8 km per second, the ground would curve away beneath it as fast as it falls, so it would keep falling forever without landing. Newton imagined precisely this with his cannonball thought experiment. So an orbit is really a projectile moving so fast that its parabola wraps right around the planet into a closed loop.

Kinetica's three worlds model a frictionless spinning table, a central-gravity field where a sun pulls objects into orbits, and an optics lab where light bends through glass. None of these is a flat field with steady downward gravity, which is what classic projectile motion needs. That is why the demo on this page is shown directly as an animated arc rather than through the game engine. Conceptually, though, projectile motion is built entirely from ideas Kinetica does model — the horizontal inertia of the table and the vertical free fall of the Field — combined into one parabola.

It was one of the first great triumphs of treating motion mathematically. For centuries people believed a cannonball flew straight and then dropped, until Galileo showed its path is a smooth parabola, by realising the horizontal and vertical motions could be separated. This insight — that a complex motion is the sum of simpler independent ones — became a template for all of physics, from planetary orbits to the components of forces. Projectile motion also had immediate practical weight, letting gunners and engineers predict where a launched object would land.

The clean 45° rule assumes the projectile is launched and lands at the same height, with no air. Real throwers break both assumptions. A shot-putter or long-jumper releases from well above the ground, which lowers the best angle to around 40° or less, because the extra launch height buys flight time a flatter, faster throw can exploit. Air resistance pushes the optimum lower still for light, fast objects, and a javelin’s aerodynamic lift changes things again. So athletes and coaches tune the angle to the real conditions — the 45° rule is the ideal starting point, not the final answer.