Projectile Motion Formulas
Ignoring air resistance, a projectile moving through a uniform gravitational field traces a parabolic path. The kinematic equations are resolved by breaking the initial velocity into x and y vectors.
Frequently Asked Questions
Assuming a flat surface and no air resistance, horizontal range R = (initial_velocity^2 * sin(2 * launch_angle)) / gravity.
Projectile motion is the curved motion of an object launched into the air that then moves under gravity alone (air resistance neglected in basic analysis). The trajectory is a parabola. The horizontal component has constant velocity (no force acts horizontally), while the vertical component undergoes constant deceleration upward and acceleration downward due to gravity (g = 9.81 m/s² on Earth's surface).
Horizontal: x = v₀ × cos(θ) × t. Vertical position: y = v₀ × sin(θ) × t − ½ × g × t². Horizontal velocity: vx = v₀ × cos(θ) (constant). Vertical velocity: vy = v₀ × sin(θ) − g × t. Maximum height: H = (v₀ × sin θ)² / (2g). Time of flight: T = 2 × v₀ × sin(θ) / g. Horizontal range: R = v₀² × sin(2θ) / g.
Range is maximised at exactly 45° (ignoring air resistance), because Range R = v₀² × sin(2θ) / g, and sin(2θ) reaches its maximum value of 1.0 when 2θ = 90°, i.e., θ = 45°. With air resistance, the optimal angle drops below 45° (typically 38–43° for sports balls). Complementary angles (e.g., 30° and 60°) give equal ranges in the no-air-resistance case.
The same equations apply with the local gravitational acceleration: Moon (g = 1.62 m/s²) — range and height are ~6× greater than Earth for same initial velocity; Mars (g = 3.72 m/s²) — range ~2.6× greater than Earth; Jupiter (g = 24.8 m/s²) — range ~2.5× less than Earth. This is important for spacecraft landing calculations, ballistics on other bodies, and science education.
Real projectile motion differs from the ideal model due to: air resistance (drag force ∝ v²) — reduces range and lowers optimal launch angle; spin (Magnus effect) — golf balls, baseballs, and soccer balls curve due to spin-induced pressure differences; wind — adds or subtracts horizontal velocity component; Coriolis effect — significant for artillery shells and long-range ballistics; and variation of g with altitude — negligible for most sports and engineering applications.