Projectile Motion Calculator - Range, Height & Time

Projectile motion formulas

• Range: R = v₀² × sin(2θ) / g
• Max height: H = v₀² × sin²(θ) / (2g)
• Time of flight: T = 2 × v₀ × sin(θ) / g
• Horizontal velocity: vₓ = v₀ × cos(θ)
• Initial vertical velocity: v_y₀ = v₀ × sin(θ)

Maximum range

Range is maximised at a launch angle of 45°. At this angle sin(2θ) = sin(90°) = 1, giving the longest horizontal distance.

Trajectory arc

The trajectory of a projectile is a parabola. The key points along the arc are:

  Max height (H)
       ●
      / \
     /   \
    /     \
   /       \
●───────────●
 Launch          Land
 (0, 0)         (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 = 9.8 m/s², reaches zero at max height H, then accelerates back to −v₀·sin(θ) at landing.
• Symmetry: the ascending and descending halves are mirror images (on flat ground with no air resistance).

Air resistance effects

Real projectiles experience aerodynamic drag proportional to velocity squared (F_drag = ½ ρ v² C_d A). Drag reduces range and lowers the optimal launch 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 45° optimum.

Real-world examples

Coordinate system note

This calculator assumes: flat ground at y = 0 (launch and landing at the same height), no wind, and uniform gravity (g = 9.80665 m/s²). For drag effects, see the Reynolds Number Calculator.