Ballistic Trajectory Calculator
Enter a launch velocity and angle to find the projectile's time of flight, maximum height, and range.
Assumes flat ground and no air resistance.
How projectile motion is calculated
This calculator uses standard kinematics equations for projectile motion under gravity alone, with no air resistance. Time of flight is 2v₀sin(θ)/g, maximum height is (v₀sin(θ))²/(2g), and range is v₀²sin(2θ)/g, where v₀ is launch velocity, θ is launch angle, and g is gravitational acceleration (9.80665 m/s²).
These equations separate the launch velocity into horizontal and vertical components using the launch angle — the vertical component determines how high and how long the projectile flies, while the horizontal component (combined with the time of flight) determines the range.
Frequently asked questions
What launch angle gives the maximum range?
For a given launch velocity on flat ground with no air resistance, 45° gives the maximum range, since sin(2θ) in the range formula reaches its maximum value of 1 when 2θ = 90°, which happens at θ = 45°.
Why does this calculator ignore air resistance?
Air resistance is complex, depending on the projectile's shape, size, speed, and air density, requiring more advanced equations to model accurately. This calculator gives the idealized, no-drag trajectory, which is a reasonable first approximation for many short-range or low-speed scenarios but increasingly inaccurate for high-speed or long-range projectiles.
Does this account for launch and landing at different heights?
No — this assumes the projectile launches and lands at the same height (flat ground). Launching from an elevated position or landing at a different height requires a more complete set of equations accounting for that height difference.