Maximum Height Calculator

The Maximum Height Calculator determines the peak altitude reached by a projectile launched at a given speed and angle. At maximum height, the vertical velocity component is zero while the horizontal component remains unchanged, making this a critical point in the trajectory analysis of any ballistic object.

The maximum height is derived from the kinematic equation $$v_y^2 = v_{0y}^2 - 2gH$$. Setting $$v_y = 0$$ at the apex yields: $$H = \frac{v_0^2 \sin^2(\theta)}{2g}$$ where $$v_0$$ is launch speed, $$\theta$$ is the launch angle, and $$g$$ is gravitational acceleration. The time to reach maximum height is half the total flight time: $$t_H = \frac{v_0 \sin(\theta)}{g}$$.

Maximum height depends on the square of the vertical velocity component ($$v_0 \sin\theta$$). A vertical launch ($$\theta = 90°$$) achieves the greatest possible height $$H_{\max} = v_0^2/(2g)$$ because all kinetic energy converts to gravitational potential energy. As the angle decreases, more energy goes into horizontal motion and less into vertical rise.

At the apex of the trajectory, the projectile has minimum speed but is not stationary (unless launched vertically). The speed at maximum height equals the horizontal component $$v_0 \cos\theta$$, which remains constant throughout the flight in the absence of air resistance. This is why a ball launched at 45° is moving at $$v_0/\sqrt{2}$$ at its highest point.

Understanding maximum height is essential in many practical scenarios: determining if a baseball clears an outfield wall, ensuring fireworks detonate at the correct altitude, calculating the apex of artillery shells for fragmentation timing, designing water fountain trajectories, and analyzing the biomechanics of jumping athletes.

The energy perspective provides additional insight: at maximum height, the kinetic energy $$\frac{1}{2}mv_0^2 \sin^2\theta$$ has converted to potential energy $$mgH$$, while the remaining kinetic energy $$\frac{1}{2}mv_0^2 \cos^2\theta$$ drives horizontal motion. This energy partition principle is fundamental to understanding all projectile trajectories.