Projectile motion describes the curved path of an object launched into the air under the influence of gravity, neglecting air resistance. The trajectory is a parabola governed by two independent components: horizontal motion at constant velocity (x = v₀cosθ × t) and vertical motion under gravitational acceleration (y = v₀sinθ × t - ½gt²). A baseball thrown at 40 m/s at 30° above horizontal reaches a maximum height of 20.4 meters, travels 141 meters horizontally, and stays airborne for 4.1 seconds. The optimal launch angle for maximum range is 45° in a vacuum, but air resistance shifts this to 30-40° for real projectiles. Our projectile motion calculator computes trajectory parameters from initial velocity and launch angle: maximum height, range, time of flight, velocity at any point, and the complete trajectory path. It handles projectiles launched from elevated positions, calculates impact velocity, and supports both metric and imperial units.
Key projectile motion equations
The four essential equations (assuming launch from ground level, no air resistance): Range R = v₀²sin(2θ)/g — maximum at θ = 45°. Maximum height H = v₀²sin²(θ)/(2g). Time of flight T = 2v₀sin(θ)/g. These decompose the launch velocity into components: v_x = v₀cos(θ) remains constant throughout flight; v_y = v₀sin(θ) - gt decreases until zero at peak, then increases downward. At any time t: horizontal position x = v₀cos(θ)t, vertical position y = v₀sin(θ)t - ½gt². Example: a cannonball fired at 100 m/s at 45°: R = 100²×sin(90°)/9.81 = 1,019 m, H = 100²×sin²(45°)/(2×9.81) = 255 m, T = 2×100×sin(45°)/9.81 = 14.4 seconds.
Launch angle optimization
In a vacuum, maximum range occurs at 45° for any launch speed. But complementary angles (e.g., 30° and 60°) produce the same range — 30° gives a flat, fast trajectory while 60° gives a high, slow one. For a launch from elevation (cliff, hill), the optimal angle shifts below 45° because the projectile has extra vertical distance to travel. From a height h, optimal angle ≈ arctan(v₀/√(v₀² + 2gh)), which approaches 45° as v₀ increases relative to h. With air resistance, optimal angle drops to 30-40° for most real projectiles because high trajectories spend more time at high altitude where air resistance decumulates. Artillery calculations must also account for wind, the Coriolis effect (Earth's rotation), and aerodynamic properties of the shell.
Real-world applications and air resistance effects
Sports applications: a basketball free throw (initial speed ~7 m/s, angle ~52°, release height ~2.4 m, hoop at 3.05 m, distance 4.6 m) has a narrow window of successful angles and speeds. A golf drive (initial speed 70 m/s, angle 10-12° with backspin) uses the Magnus effect (spin-induced lift) to achieve ranges of 250+ meters — far exceeding the vacuum prediction at that angle. Air resistance (drag force Fd = ½ρv²CdA) significantly affects trajectory: a baseball hit at 45 m/s at 35° travels approximately 120 meters in a vacuum but only 95-100 meters with air resistance — a 20% reduction. Drag force increases with the square of velocity, so faster projectiles are proportionally more affected. Terminal velocity occurs when drag equals gravity: approximately 30 m/s for a baseball, 55 m/s for a skydiver (spread position), and 90 m/s for a skydiver (head-down).