Projectile Range Calculator
Enter a launch speed and angle to get the horizontal range on level ground — and see why a 45° launch carries the farthest.
Range from speed and angle
Enter the launch speed in metres per second and the angle in degrees and the projectile range calculator returns the horizontal distance (R = v² × sin(2θ) / g) in metres.
Level ground, no drag
The formula assumes flat ground, the same launch and landing height, and no air resistance — an idealised but very useful first estimate.
What is projectile range?
How far a launched object travels
The projectile range calculator works out how far an object travels horizontally before it lands when it is launched at an angle over flat ground. Any object thrown, kicked, or fired follows a curved path, and the horizontal distance it covers depends on just two things you control: the launch speed and the launch angle. With gravity pulling it down at a constant rate, the range follows a clean relationship, R = v² × sin(2θ) / g, that turns a speed in metres per second and an angle in degrees into a distance in metres. It is the number behind the reach of a water jet, the carry of a struck ball, and the textbook study of projectile motion.
Enter a launch speed in metres per second and an angle in degrees to get the horizontal range on level ground instantly.
The range is the launch speed squared, multiplied by the sine of twice the launch angle, divided by gravity (g = 9.80665 m/s²).
R = v² × sin(2θ) / gThe angle enters through sin(2θ), which peaks at 2θ = 90° — that is, at a launch angle of 45°. The speed enters squared, so the range grows with the square of the launch speed: double the speed and the range quadruples. Use metres per second for the speed and degrees for the angle and the range comes back in metres.
Suppose an object is launched at 20 m/s at an angle of 45° over level ground.
Double the angle and take the sine
2 × 45° = 90°, and sin(90°) = 1 — the largest the sine term can be.
Square the launch speed
20² = 400 — the squared speed that scales the range.
Divide by gravity
400 × 1 ÷ 9.80665 = 40.7886 m — the horizontal range on level ground.
The range tells you how far the object lands from where it left the ground, measured along the flat surface. Two features of the formula are worth knowing. First, the maximum range for a given speed always happens at a launch angle of 45°, because sin(2θ) reaches its peak of 1 there — steeper or shallower launches both fall short. Second, the range is symmetric about that 45° peak: complementary angles give exactly the same range, so a 30° launch and a 60° launch carry an object the same distance (the 60° shot simply rises higher and stays aloft longer to cover it). The launch speed is the bigger lever overall, since the range grows with its square: a 30° throw at 28 m/s reaches farther than a 45° throw at 20 m/s. Use these rules to read your result — if you want maximum distance, aim near 45°; if a steeper or flatter trajectory matters, pick the complementary angle that gives the same reach.
The formula is exact for the ideal case, but a few assumptions are baked in.
No air resistance, level ground, equal heights
This calculator ignores air resistance, which in reality shortens the range and lowers the optimal angle below 45°, especially for light or fast objects. It also assumes the launch and landing points sit at the same height on flat, level ground — launching from a cliff or into a valley changes the result. Treat the answer as a clean upper-bound estimate for a vacuum-like case, not a precise prediction for a windy day.