Range, peak height, time of flight, impact speed and angle, with full SVG trajectory.
Vacuum model: ignores air drag, Magnus effect, and Coriolis. Range is maximized at 45° on flat ground; with non-zero launch height the optimal angle is below 45°.
Projectile motion is the canonical introduction to two-dimensional kinematics in every introductory physics course: a body launched at an angle into a uniform gravitational field, ignoring air drag. The math is elegant — the trajectory is a parabola, the range is maximized at 45° on flat ground, and the time of flight is purely geometric. Beyond physics class, the same equations underlie water-fountain design, basketball shooting biomechanics, ballistics, fireworks display planning, and the trajectory of a thrown frisbee. This calculator solves the full problem (range, peak height, time of flight, impact velocity, impact angle), draws the parabolic trajectory at 80-point resolution with peak and impact markers, and supports an arbitrary launch height — useful when modeling cliff-edge throws or basketball shots where the ball doesn't start at ground level.
Initial velocity is decomposed into horizontal and vertical components: vₓ = v₀ · cos θ; vᵧ = v₀ · sin θ.
Time to peak = vᵧ / g (vertical velocity decays linearly under gravity).
Peak height above launch = vᵧ² / (2 · g). With launch height h₀: peak_total = h₀ + vᵧ² / (2g).
Time of flight (until projectile lands at y = 0): solve h₀ + vᵧ · t − ½ · g · t² = 0:
t_flight = (vᵧ + √(vᵧ² + 2 · g · h₀)) / g
Range = vₓ · t_flight.
Impact velocity components: vᵢₓ = vₓ (no horizontal drag in vacuum); vᵢᵧ = vᵧ − g · t_flight.
Impact speed = √(vᵢₓ² + vᵢᵧ²).
Impact angle below horizontal = atan(|vᵢᵧ| / vᵢₓ).
The chart samples the trajectory at 80 points using y(t) = h₀ + vᵧ · t − ½ · g · t², plots the parabola in red, and marks the peak (top of arc) and impact (right end). The Y-axis padding adds 10 % above the peak and 5 % to the right of impact for visual clarity.
Enter the initial velocity in m/s (typical 10–50 for thrown objects, 200+ for fired projectiles). Enter the launch angle in degrees from horizontal (0 = pure horizontal, 90 = straight up). Enter the launch height above ground in meters (0 for ground-level launches). Enter the gravity (default 9.81 m/s² on Earth; use 1.62 for Moon, 3.71 for Mars). The result panel shows range, peak height, time of flight, time to peak, impact speed, and impact angle, plus the trajectory curve.
A baseball thrown at 25 m/s at 45° from launch height 1.5 m on Earth:
A water-fountain jet at 8 m/s, 70°:
A basketball shot from 2.0 m height at 8 m/s, 50°: - vₓ = 5.14 m/s. vᵧ = 6.13 m/s. Time of flight: 1.54 s. Range: 7.94 m.
Air drag matters past ~ 30 m/s. The vacuum model overestimates range by 5 % at 50 m/s, by 30 % at 200 m/s, by an order of magnitude at 800 m/s. Real ballistics needs a drag model.
Wind. Cross-wind moves the impact horizontally; head/tail wind shortens or extends range. Not modeled.
Spin and Magnus effect. A spinning baseball curves due to the Magnus force; a topspin tennis ball drops faster than a flat one. Not modeled.
Coriolis force at long ranges. Earth's rotation deflects projectiles measurably past ~ 1 km range. Negligible for sports, critical for artillery.
Launch height affects optimal angle. The 45° rule is for launch from ground level. With h₀ > 0, the optimal angle for maximum range is below 45° (45° − arctan(h₀ / range_estimate) / 2 approximately). The calc lets you enter both, and you can sweep the angle to find optimum.
Variable gravity. g varies by 0.5 % between equator and pole, and by altitude. For everyday problems, g = 9.81 is fine.
Aerodynamic-shape dependence. A javelin and a shot put behave very differently because of drag and lift. The calc treats all projectiles as spherical-with-no-drag.
Earth's curvature. For range past ~ 30 km, the earth curves away below the projectile, extending the effective range. Negligible at sports scale.
Angle conventions. The calc uses degrees from horizontal. Some textbooks use degrees from vertical (zenith angle) — convert if needed.
Negative launch heights. The calc supports h₀ ≥ 0 (above ground). Below-ground launches (a thrown ball going into a pit) need the formula extended.
Energy at impact. Impact kinetic energy = ½ · m · v_impact² — useful for sizing landing damage. The calc doesn't compute energy (mass not entered).