Projectile Motion Calculator

Projectile motion describes the path of an object launched into the air under gravity alone — no air resistance in the ideal model. Baseballs, artillery shells, fountain jets, and long-jump athletes all follow approximately parabolic trajectories. Splitting motion into horizontal and vertical components turns a curved path into two independent straight-line problems governed by constant horizontal velocity and constant vertical acceleration.

Key relationships (launch from ground level, angle θ, speed v, gravity g):

Horizontal velocity: v_x = v cos θ

Vertical velocity: v_y = v sin θ

Max height: h = (v² sin² θ) / (2g)

Range: R = v² sin(2θ) / g

Flight time: t = 2v sin θ / g

Where θ is launch angle from horizontal, v is initial speed, and g ≈ 9.81 m/s². Maximum range on level ground occurs at θ = 45° for a given speed. Launch and landing at different elevations shift results — this calculator assumes level ground unless height offset is provided.

Worked example: A ball kicked at 20 m/s and 35°. v_x = 20 cos 35° ≈ 16.38 m/s, v_y = 20 sin 35° ≈ 11.47 m/s. Max height h = 11.47² / (2 × 9.81) ≈ 6.7 m. Flight time t = 2 × 11.47 / 9.81 ≈ 2.34 s. Range R = 16.38 × 2.34 ≈ 38.3 m. At 45° with same speed, range would be 20² sin 90° / 9.81 ≈ 40.8 m.

Compare with speed-distance-time for horizontal segments, kinetic energy for launch energy, and acceleration for vertical g. Real trajectories need drag corrections at high speed — golf balls and artillery deviate noticeably from the ideal parabola.