Projectile Motion Calculator - Range, Height & Time

How to use this tool

1. Enter the initial speed, launch angle, and optional initial height; use 0 for a ground-level launch.
2. Gravity defaults to 9.80665 m/s² (or 32.174 ft/s² in imperial mode); change it if needed.
3. Results update instantly.

Projectile motion formulas

• Horizontal velocity: vₓ = v₀ × cos(θ)
• Initial vertical velocity: v_y₀ = v₀ × sin(θ)
• Time to ground: T = [v_y₀ + √(v_y₀² + 2gh₀)] / g
• Range: R = vₓ × T
• Maximum height above ground: H = h₀ + v_y₀² / (2g)
• Time to peak: t_peak = v_y₀ / g
• When h₀ = 0: T = 2v_y₀ / g and H = v_y₀² / (2g)

Maximum range

With no air resistance and launch and landing at the same height, a 45° launch angle gives maximum range because sin(2θ) reaches its largest value at 90°. If the projectile starts above its landing point, the ideal no-drag angle for maximum range is below 45°; the result depends on the launch height as well as speed and gravity.

Trajectory arc

The trajectory is a parabola. This diagram shows a launch from height h₀ that lands at ground level, y = 0; when h₀ is zero, the launch and landing points are level.

  Max height (H)
       ●
      / \
     /   \
    /     \
   /       \
   ●───────────●
 Launch          Land
 (0, h₀)        (R, 0)

←──── Range (R) ────->

• Horizontal component: constant velocity v₀·cos(θ) throughout the flight (ignoring air resistance).
• Vertical component: starts at v₀·sin(θ), decelerates at g = 9.80665 m/s², reaches zero at max height H, then accelerates downward. With h₀ above ground, the impact vertical speed is −√(v_y₀² + 2gh₀), so it is not necessarily −v_y₀.
• Symmetry: the ascending and descending halves are mirror images only when launch and landing heights match and air resistance is ignored.

Air resistance effects

Real projectiles experience aerodynamic drag proportional to velocity squared (F_drag = ½ ρ v² C_d A). Drag reduces range and shifts the ideal no-drag optimum; for a same-height launch it often moves the best angle below 45°. For example, Olympic shot put athletes throw at approximately 42° rather than 45° because air resistance acts more strongly at the higher velocities associated with steeper angles. The heavier the projectile and the lower its drag coefficient, the closer real-world behavior approaches the ideal no-drag result.

Real-world examples

Coordinate system note

Ground level is y = 0, and the initial height h₀ is measured above that ground. The calculator finds when the projectile reaches y = 0, so a positive h₀ represents a launch from a table, platform, or cliff; at h₀ = 0, launch and landing heights match. It assumes no wind and uniform gravity, defaulting to 9.80665 m/s² (32.174 ft/s² in imperial mode). Air resistance is not included; for drag-related concepts, see the Reynolds Number Calculator.