Projectile Motion Calculator
Compute the range, peak height, flight time and impact speed of a projectile launched at a given speed, angle and height.
m/s
°
m
Horizontal distance to impact
- 1
Horizontal component vx = v₀ × cos θ
20 × cos(45°) = 14.142 - 2
Vertical component vy = v₀ × sin θ
20 × sin(45°) = 14.142 - 3
Time of flight T
(14.142 + √(14.142² + 2 × 9.81 × 0)) ÷ 9.81 = 2.883Solve the vertical equation of motion for when the projectile reaches y = 0. - 4
Range R = vx × T
14.142 × 2.883 = 40.77
How does this calculator work?
For a projectile launched at speed v₀ and angle θ from height h₀, split velocity into vx and vy, then time of flight T = (vy + √(vy² + 2gh₀))/g, maximum height H = h₀ + vy²/(2g), and range R = vx·T, using g = 9.81 m/s² and ignoring air resistance.
Formula
How this is calculated
Enter the launch speed v₀, the launch angle θ measured above the horizontal, and the launch height h₀. The speed is split into horizontal and vertical components vx = v₀·cos θ and vy = v₀·sin θ. Gravitational acceleration is taken as g = 9.81 m/s².
The time of flight comes from solving the vertical equation of motion for when the projectile returns to ground level (y = 0): T = (vy + √(vy² + 2g·h₀)) / g. The maximum height is reached when the vertical velocity is zero, giving H = h₀ + vy²/(2g). Multiplying horizontal velocity by total time gives the range R = vx·T. The impact speed combines the constant horizontal component with the final vertical velocity (vy − gT): √(vx² + (vy − gT)²).
The model assumes no air resistance, a flat landing surface at y = 0, and constant gravity. Speed and height must be non-negative. At θ = 90° the projectile goes straight up so the range is zero, and launching from a positive height extends both flight time and range compared with a ground-level launch.
Frequently asked questions
For a launch from ground level (h₀ = 0), 45° maximizes range. When launching from a positive height, the optimal angle is slightly below 45°.
No. This calculator uses idealized projectile motion with no drag, so real-world ranges for light or fast objects will be shorter.
When launching from a height above the landing surface, gravity adds vertical speed during the extra descent, so the projectile can strike faster than it left.
Also known as
TG we-Calculate Editorial Team. (2026). Projectile Motion Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/projectile-motion-calculator
TG we-Calculate Editorial Team. "Projectile Motion Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/projectile-motion-calculator.
TG we-Calculate Editorial Team, "Projectile Motion Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/projectile-motion-calculator
@misc{wecalculate_projectile_motion_calculator, title = {Projectile Motion Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/projectile-motion-calculator}}, year = {2026}, note = {TG we-Calculate} }
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