Projectile Motion Calculator

Enter initial velocity, launch angle, and optionally initial height. Calculates range, maximum height, time of flight, and velocity components.

Launch Parameters

Results

Range
Max Height
Time of Flight
Horizontal vₓ
Vertical v₀ᵧ
Impact Speed

Assumes flat ground and no air resistance. Range shown is horizontal distance from launch point to landing.

Formulas Used

Velocity Components
vₓ = v₀ × cos(θ)
v₀ᵧ = v₀ × sin(θ)
Range (flat ground)
R = v₀² × sin(2θ) / g
(only when h₀ = 0)
Maximum Height
h_max = h₀ + v₀ᵧ² / (2g)
Time of Flight
T = (v₀ᵧ + √(v₀ᵧ² + 2gh₀)) / g

Quick Examples

Angle Quick Reference

30°Short, fast
45°Max range
60°High, short
90°Straight up
Note: 30° and 60° have the same range at equal initial speed on flat ground.

Related Calculators

→ Kinetic Energy Calculator → Newton's Second Law → All Science Calculators

Frequently Asked Questions

Why does 45° give maximum range?

The range formula R = v₀²sin(2θ)/g is maximized when sin(2θ) = 1, which occurs at 2θ = 90°, so θ = 45°. This balances horizontal velocity with time of flight optimally.

Does this account for air resistance?

No. This calculator uses ideal projectile motion equations that assume no air resistance. Real-world trajectories differ, especially at high speeds or for objects with high drag coefficients.

What is initial height h₀?

Initial height is the vertical distance above the landing surface at launch. For example, throwing a ball off a 10 m cliff has h₀ = 10 m. Leave blank or enter 0 for a ground-level launch.

What is impact speed?

Impact speed is the magnitude of the velocity vector when the projectile hits the ground. It combines both horizontal (vₓ) and vertical (vᵧ at impact) components using the Pythagorean theorem: v = √(vₓ² + vᵧ²).