Projectile Motion: Range, Angle and Parabola Explained

Two motions that do not disturb each other

Hold one ball over the edge of a table and flick another sideways at the same instant. Which hits the floor first? Neither: they land together. Gravity pulls both down in the same way, and a sideways motion does not change that. This is the key idea of the chapter: horizontal and vertical motions are independent. Horizontally the ball keeps a steady speed; vertically it falls with s = g·t² / 2. For a table 1.25 m high, t = √(2 × 1.25 / 9.81) ≈ 0.505 s. A ball thrown at 4 m/s lands 4 × 0.505 ≈ 2.02 m from the edge.

The path is a parabola

Join the two motions and the ball draws a curve called a parabola. Launched with speed v at angle θ (theta), the ball has a horizontal speed v·cos θ and a starting vertical speed v·sin θ. After time t its position is x = v·cos θ·t and y = v·sin θ·t − g·t² / 2. Eliminate t and y depends on x squared: that is the equation of a parabola. Galileo showed this in his book of 1638. The ball rises, slows at the top where its vertical speed is zero, then falls faster and faster.

Numbers for a 20 m/s throw

Take v = 20 m/s and θ = 45°. Then v·sin θ = 20 × 0.7071 ≈ 14.1 m/s. The ball reaches its top when the vertical speed has dropped to zero, at the height H = (v·sin θ)² / (2g) = 200 / 19.62 ≈ 10.2 m. The whole flight lasts T = 2·v·sin θ / g = 2 × 14.14 / 9.81 ≈ 2.88 s. In the sandbox readouts you will find these same three kinds of numbers, the height, the time and the range, and they change as you move the sliders.

Why 45° goes farthest

With the angle you trade height for distance. Throw straight up and the ball returns to your hand; throw almost flat and it drops quickly. On level ground without air the range is R = v²·sin 2θ / g. The factor sin 2θ is biggest, equal to 1, when 2θ = 90°, that is, when θ = 45°. With v = 20 m/s: R = 400 × 1 / 9.81 ≈ 40.8 m. Because the range depends on v squared, doubling the throw speed gives four times the distance. In the sandbox, sweep the angle from 10° to 80° and see where the landing point is farthest.

Two angles, one landing spot

Here is a surprise: 30° and 60° land at the same place. Their sum is 90°, and sin 60° and sin 120° are both 0.866, so the formula gives the same range: R = 400 × 0.866 / 9.81 ≈ 35.3 m for v = 20 m/s. The paths are very different. At 30° the ball is up for 2 × 20 × 0.5 / 9.81 ≈ 2.04 s and peaks at 5.1 m. At 60° it hangs for about 3.53 s and climbs to 15.3 m. Any pair of angles that add up to 90° behaves this way, such as 20° and 70°, or 40° and 50°.

What air changes

All of this assumed an empty world. Real air pushes against a flying ball, and the harder the faster it goes. The ball loses speed all the way, so the path is no longer a perfect parabola: it rises steeply and comes down even steeper. The range is shorter than R = v²·sin 2θ / g predicts, and the best angle drops below 45°, because a lower throw spends less time fighting the air. Archers, golfers and gunners all learned this. Switch the air on in the sandbox, keep the speed fixed, and compare the range at 45° with air on and off.

Projectile motion