About the Maximum Height Calculator – Projectile Motion
When a projectile is launched with an initial speed v0 at an angle θ above the horizontal, gravity constantly decelerates its upward motion until, for an instant, its vertical velocity reaches zero — the apex, or maximum height. Only the vertical component of the initial velocity, v0y = v0 sinθ, affects how high the projectile climbs; the horizontal component (assuming no air resistance) does not change the height at all. This calculator applies the standard kinematic formula for maximum height to your launch speed, angle, and starting height.
Deriving the maximum height formula
Starting from the vertical motion equation v² = v0y² − 2g·h and setting the vertical velocity v to zero at the apex gives 0 = v0y² − 2gH, which rearranges to H = v0y² / (2g) = (v0² sin²θ) / (2g). This is the height gained above the launch point. If the projectile launches from a platform or elevation h0 above the ground, the height above the ground is simply H + h0. The time to reach the apex follows from v = v0y − g·t with v = 0, giving t = v0y / g = (v0 sinθ) / g.
Getting accurate results
- Measure the launch angle from the horizontal (0° = straight sideways, 90° = straight up), not from the vertical.
- This formula assumes no air resistance and constant gravitational acceleration — accurate for dense, compact projectiles over short ranges (a thrown ball, a launched rocket in the first seconds), less accurate for light or highly aerodynamic objects (a badminton shuttlecock, a beach ball) where drag noticeably reduces the real peak height.
- Keep velocity and height units consistent with the selected unit system; mixing m/s with feet (or vice versa) will silently produce a wrong answer.
- A launch angle of 90° maximizes height for a given speed, but a launch angle of 45° maximizes horizontal range (for launch and landing at the same elevation) — they are not the same optimization.