Projectile Motion

A projectile is an object moving under gravity alone after being launched. The whole subject rests on one idea.

The components are independent — horizontal and vertical motion do not affect each other. Gravity acts vertically, so it changes the vertical velocity and leaves the horizontal velocity completely alone.

Example: a bullet fired horizontally and a bullet dropped from the same height at the same moment hit the ground together. The fired bullet travels much further, but its vertical motion is identical to the dropped one, because gravity does not care how fast it is moving sideways.

Horizontal motion — no horizontal force acts once launched (ignoring air resistance), so the horizontal velocity is constant:

x = v t

Vertical motion — a constant acceleration of g downwards, so the suvat equations apply with a = g.

The method — resolve into components, treat the two directions separately, and use time as the only quantity they share. Time is the link: find it from one direction and use it in the other.

Example: a ball thrown horizontally at 15 m s⁻¹ from a cliff 20 m high. Vertically, 20 = ½ × 9.81 × t², so t = 2.02 s. Horizontally, x = 15 × 2.02 = 30 m. Notice the horizontal speed played no part in finding the time.

Launched at an angle — resolve the launch velocity first. For speed u at angle θ to the horizontal, the horizontal component is u cosθ and the vertical component is u sinθ. Then proceed exactly as before.

Symmetry — for a projectile landing at its launch height, the path is symmetric: the time up equals the time down, and the landing speed equals the launch speed. At the highest point the vertical velocity is zero while the horizontal velocity is unchanged, which is why a projectile at the top of its arc is still moving.

What air resistance changes — it reduces both the range and the maximum height, and destroys the symmetry: the descent is steeper than the ascent. Every calculation here assumes it is negligible, which is reasonable for dense compact objects at modest speeds and poor for a shuttlecock.