Gravity alone: the parabola you get with no air

Every ball-flight story starts with the simplest case: a ball moving through empty space, with gravity as the only force. The key idea is that the two directions do not disturb each other.

• Horizontally nothing pushes the ball, so its horizontal speed never changes.
• Vertically gravity changes the vertical speed by 9.81 m/s every second (g ≈ 9.81 m/s²): it slows the ball on the way up and speeds it up on the way down.

Join the two motions and the path is a parabola.

Launch a ball at speed v and angle θ above the ground. Split the speed into parts: vₓ = v cos θ (sideways) and v_y = v sin θ (upwards). The ball stays up for t = 2v sin θ / g, and on level ground it lands at the range

R = v² sin 2θ / g

Because sin 2θ is largest when 2θ = 90°, the longest range is at 45°. Angles that add up to 90° (30° and 60°, say) give the same range, but the steeper one goes higher and stays in the air longer.

Worked example. A kick at 30 m/s and 20° above the ground. The sideways speed is 30 cos 20° ≈ 28.2 m/s, and the upward speed is 30 sin 20° ≈ 10.3 m/s. The ball is airborne for about 2.1 s and rises about 5.4 m. The range is

R = 30² × sin 40° / 9.81 ≈ 59 m.

This is the no-air answer. The next lesson shows how much of it survives in real air.

For a deeper look at frictionless motion, collisions and ramps, see the Projectile world in the Physics Lab. Here, it is just our starting line.