Projectile Range Calculator
Compute projectile horizontal range from v₀, angle and initial height.
Alcance ≈ — m
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°.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.