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.
Projectile Motion: From First Principles
A projectile is any object launched into the air and subsequently moving only under the influence of gravity (and, in idealised analysis, no air resistance). The classical analysis decomposes the motion into two independent components: horizontal motion at constant velocity, and vertical motion under constant downward acceleration g = 9.81 m/s². This decomposition is the trick that makes the equations tractable: x and y motions are coupled only through time, not through forces.
The Equations of Motion
For a projectile launched from ground level with initial speed v₀ at angle θ above the horizontal: horizontal velocity vx = v₀ cosθ (constant); vertical velocity vy(t) = v₀ sinθ − gt; horizontal position x(t) = v₀ cosθ · t; vertical position y(t) = v₀ sinθ · t − ½gt². From these, the maximum height is reached when vy = 0, giving H = v₀² sin²θ / (2g); the time of flight (back to launch height) is T = 2v₀ sinθ / g; and the horizontal range is R = v₀² sin(2θ) / g. The factor sin(2θ) is maximum at θ = 45°, which is why on level ground the maximum-range angle is 45°. The trajectory itself is a parabola y(x) = x tanθ − gx² / (2v₀² cos²θ).
Worked Example: Punter’s Optimum Angle
An American football punter kicks a ball at 25 m/s. What angle gives the maximum hang time, and what angle gives the maximum range? Hang time T = 2v₀ sinθ / g is maximised at θ = 90° (straight up; T = 2 × 25 / 9.81 = 5.10 s — but R = 0, useless). Range R = v₀² sin(2θ) / g peaks at θ = 45° (R = 625 / 9.81 = 63.7 m, with T = 2 × 25 × 0.707 / 9.81 = 3.60 s). Punters compromise around 50–55° to trade some range for hang time, giving the coverage team time to get downfield. The projectile calculator lets you sweep angles and read off range, height, and hang time simultaneously.
Launch From a Height: The Asymmetric Case
If the launch height differs from the landing height (a cliff jumper, a cannonball fired from a ramp, a basketball shot to a higher hoop), the symmetric formulae do not apply. The full solution requires solving the quadratic y(t) = hlaunch + v₀ sinθ · t − ½gt² = hlanding for t. The optimal launch angle for maximum range from a height h above the landing point is θopt = arctan(v₀ / √(v₀² + 2gh)), which is always less than 45° when h > 0 and greater than 45° when launching uphill. The calculator handles both cases when you specify a non-zero launch height.
Air Resistance: Where the Idealisation Breaks
Real projectiles experience drag FD = ½ ρ v² CD A. For a typical baseball at major-league speeds (~40 m/s), drag halves the actual range compared to the vacuum prediction. For a steel artillery shell at hundreds of m/s, drag is the dominant force during much of the trajectory and the parabola becomes asymmetric (steeper on the descending arm). The projectile calculator on this page is the vacuum (no-drag) solution and is best used for educational analysis, light objects at modest speeds, or first-pass estimates. For ballistics-grade work, use a numerical integrator with a velocity-dependent drag coefficient.
Where Projectile Motion Calculations Help
Beyond physics homework, projectile-motion calculations are used by sports analysts (golf trajectory, basketball arc, football kicking), forensic investigators (bullet trajectory analysis, blood-spatter pattern analysis), military applications (mortar and artillery sighting), entertainment engineering (theme-park ride trajectory design), and even firefighting (water-jet reach for a given pressure). The vacuum equations are the starting point for all of these; real-world tools layer drag, wind, and Coriolis corrections on top.
Related calculators: Kinetic energy · Simple pendulum · All physics calculators