Maximum height

Beginner

Highest point reached by a projectile.

Key relationH = u² sin²θ/2g

The height at the top is H = u²sin²θ/(2g), where the vertical velocity is zero. It increases with the square of the vertical component.

Worked example

u = 30 m/s at 45°: H = 900·0.5/19.6 ≈ 23 m.

Common mistake

At the top the speed is u cosθ, not zero.

In more detail

The maximum height of a projectile is H = u² sin² θ / 2g. It depends on the vertical part of the launch velocity only.

Estimating how high a ball thrown at an angle will rise.
Designing fountains and water jets to reach a target height.
A high jump coach estimates how high an athlete's centre of mass can rise from a given take-off speed and angle.

Which launch angle gives the greatest height?

Straight up, 90 degrees, gives the greatest height for a given speed.

Does the horizontal velocity affect height?

No. It only affects horizontal distance and time, not the maximum height.

Does maximum height depend on mass?

No. Ignoring air resistance, the height depends only on the launch speed and angle, not on the mass of the projectile.