Projectile motion describes anything thrown, kicked, or launched that then flies freely under gravity. A football, a stone, and a water jet all follow the same curved path, called a parabola, when air resistance is ignored. The motion splits neatly into two parts: horizontal motion at a steady speed, because nothing pushes sideways, and vertical motion that slows, stops, and falls back under gravity. This split is the key to every projectile calculation.
Two inputs decide the whole flight: launch speed and launch angle. The speed splits into a horizontal part, speed times the cosine of the angle, and a vertical part, speed times the sine of the angle. A 45-degree launch splits the speed equally, which is why 45 degrees gives the longest range on flat ground in this ideal model. A lower angle gives a flatter, faster flight, while a higher angle gives a taller, longer-lasting flight that lands closer.
Here is a worked example. Launch a ball at 20 metres per second at 45 degrees. The horizontal and vertical parts are each 20 times sine or cosine of 45 degrees, about 14.14 metres per second. Flight time is twice the vertical speed divided by gravity: 2 times 14.14 divided by 9.81, about 2.88 seconds. Maximum height is vertical speed squared divided by twice gravity: 200 divided by 19.62, about 10.19 metres. Range is horizontal speed times flight time: 14.14 times 2.88, about 40.77 metres.
Angle matters in a way that surprises many people. At 30 degrees and at 60 degrees, the same launch speed gives the same range, because the sine of twice the angle is the same for both. The difference is the shape: the 30-degree flight is low and quick, while the 60-degree flight is high and slow. Athletes choose between them by need. A shot-putter wants height to gain distance in the air, while a footballer passing low wants the ball to arrive fast and flat.
Maximum height depends only on the vertical part of the speed. Doubling the vertical speed quadruples the height, because height uses speed squared. That is why a small increase in launch angle near vertical gives a big gain in height but costs range. Flight time also depends only on the vertical part: the ball lands when its vertical journey, up and then down, is complete. Horizontal speed only decides how far it travels during that time.
The model has honest limits. Real projectiles meet air resistance, which grows with speed and shrinks range, especially for light balls like table tennis balls or shuttlecocks. Wind pushes the path sideways or shortens it, spin curves the ball, and launching from a cliff or a hill changes the landing height. For a heavy, slow ball over a short distance, the ideal answers are close. For a golf ball driven 250 metres, air resistance changes the result by a large amount.
You can test this at home with safe, simple experiments. Throw a soft ball at a marked angle, or use a garden hose at a fixed pressure and measure where the water lands at different nozzle angles. You should see the range peak near 45 degrees and fall off on both sides. A phone slow-motion video lets you count flight time and check it against the tool. The match will not be perfect, and the gap itself teaches how much air drag matters.
To use this tool well, keep your units consistent: metres per second for speed, degrees for angle, and metres per second squared for gravity. Remember that range assumes launch and landing at the same height; if you throw from a height, the real range will be longer. Try angles in pairs, such as 30 and 60, to see the equal-range surprise for yourself, and try the Moon gravity value of 1.62 to see how far the same throw would go there.