Projectile Range Calculator

Parabolic trajectory: full equations of motion

An oblique launch under gravity (without air resistance) follows a parabolic path described by parametric equations: x(t) = v₀·cos θ·t and y(t) = v₀·sin θ·t − (1/2)·g·t². Eliminating time yields the cartesian form y(x) = x·tan θ − g·x² / (2·v₀²·cos²θ). Without drag, the angle that maximizes horizontal range on flat ground is θ = 45°; with real air resistance, the optimal angle drops to about 35° because drag grows with v². Example: v₀ = 20 m/s at θ = 45° produces a parabola reaching h_max ≈ 10.2 m at the midpoint, with horizontal range ≈ 40.8 m and total flight time ≈ 2.9 s. The trajectory is symmetric about the apex only when launch and landing heights are equal.

Applications

Sports ballistics (football free kicks, javelin and shot put, basketball arc), physics-based games (Angry Birds, Worms, Scorched Earth), defense (artillery, mortar tables), aeronautics (bomb release, drop calculations), and educational simulators in physics and biomechanics.

FAQ

Why is the path a parabola? Because horizontal velocity is constant and vertical position is quadratic in t (uniformly accelerated motion). Eliminating t between the two equations gives a polynomial of degree 2 in x.

Does mass affect the trajectory? In vacuum, no — gravity accelerates every body equally. With air drag, heavier objects of the same shape decelerate less and fly farther.

How does launch height (h₀ > 0) change the result? A non-zero starting height makes the parabola asymmetric: the descent is longer than the ascent, and the optimal angle for maximum range drops below 45°.