How this calculation works
The Projectile Motion & Trajectory Calculator determines the maximum apex height (H), total horizontal range (R), flight duration hang time (t), and impact velocity vector for objects launched in a gravitational field.
Mathematical formula and logic
v_0x = v₀·cos(θ), v_0y = v₀·sin(θ). Apex Height = h₀ + (v_0y² / (2g)). Range = v_0x × Total Flight Time.
Worked example
Launching a projectile at 30 m/s at a 45° angle from ground level (0m): Apex height is 22.94 meters, total flight hang time is 4.33 seconds, and total horizontal range is 91.74 meters (301 feet).
Calculation assumptions
- Standard Earth gravity g = 9.80665 m/s².
- Assumes zero air resistance (ideal ballistic parabolic trajectory).
Frequently asked questions
What launch angle achieves maximum horizontal range on flat ground?
Without air resistance, a 45-degree launch angle achieves the absolute maximum horizontal distance on level ground.
How does air resistance affect projectile trajectory in the real world?
Air drag flattens the trajectory, shortens the horizontal range, lowers the apex height, and shifts the optimal launch angle for maximum distance down to between 35° and 40°.
Is the vertical velocity of a projectile zero at its highest point?
Yes, at the apex peak of the trajectory, the vertical velocity component (v_y) is momentarily 0 m/s, while horizontal velocity (v_x) remains constant.
Why are horizontal and vertical motions treated independently?
Gravity acts strictly in the vertical direction (downward y-axis), leaving horizontal velocity unaffected in the absence of air drag.
How does initial launch height change the optimal angle?
Launching from an elevated cliff or platform (h > 0) means the optimal launch angle for maximum distance is less than 45 degrees.
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