Why does a projectile follow a parabola?

A thrown ball, a long jump, water from a hose — they all trace the same curved arc. Why is that curve specifically a parabola?

Split the motion in two. Sideways: nothing pushes or pulls (ignoring air), so the ball keeps a steady horizontal speed. Downwards: gravity adds the same bit of downward speed every second.

Steady sideways progress plus ever-faster falling means the path droops more and more sharply as it goes — equal horizontal steps, growing vertical drops. That precise combination is what a parabola is.

With initial speed u at angle θ: x(t) = (u cos θ)t and y(t) = (u sin θ)t − ½gt². Eliminate t by substituting t = x/(u cos θ):

y = x tan θ − g x² / (2u² cos²θ). This is y = Ax − Bx² — a quadratic in x, i.e. a parabola opening downward. Range on level ground: R = u² sin 2θ / g, maximised at θ = 45°.

Kick a ball at u = 20 m/s, θ = 45°, g ≈ 10 m/s². Time of flight: t = 2u sin θ/g = 2·20·(√2/2)/10 ≈ 2.83 s. Range: R = u² sin 90°/g = 400/10 = 40 m. Maximum height: (u sin θ)²/2g = 200/20 = 10 m. The board plots the actual curve from these equations — it peaks exactly midway, at x = 20 m.

• Thinking the horizontal speed decreases during flight — without air resistance it doesn't; only the vertical component changes.
• Believing the velocity is zero at the top: only the vertical component is zero there; the ball still moves horizontally at u cos θ.
• Forgetting air resistance is being ignored: real long-range projectiles fall short of the ideal parabola and the range formula.

• NCERT Class 11 Physics, Ch. 3 — Motion in a Plane
• Khan Academy — Projectile motion