Optimal launch angle calculator

When you throw or fire a projectile from ground level onto flat ground, there is a single launch angle that maximises how far it travels before landing again. Under ideal vacuum kinematics with constant gravity, that angle is always forty-five degrees above the horizontal, and the maximum range simplifies to launch speed squared divided by g. This calculator takes launch speed and an optional gravity value and returns that optimal angle in degrees and radians, the maximum horizontal range, and the matching flight time and peak height at that angle. Use it to size a trebuchet throw, check sports or classroom estimates, or set game-physics targets before adding drag.

Why forty-five degrees maximises range on level ground

Ideal projectile motion treats the horizontal velocity as constant and the vertical velocity as linearly decreasing under a fixed downward acceleration. The projectile returns to launch height when the vertical displacement integral closes, which produces the compact range formula R equals v squared times the sine of twice the launch angle, all divided by g. The sine function reaches its maximum value of one when its argument is ninety degrees, so twice the launch angle equals ninety degrees and the optimal elevation is exactly forty-five degrees.

At that angle the horizontal and vertical launch components are equal, each equal to v over the square root of two, and the range collapses to the simple expression R_max equals v squared over g. Complementary angles such as thirty and sixty degrees share the same range under ideal conditions, but neither beats forty-five degrees on flat ground. This derivation assumes launch and land at the same height, no air resistance, no wind, and no Earth curvature.

If the landing surface is lower than the launch point, the optimal angle drops below forty-five degrees; if it is higher, the optimum rises. Always treat the forty-five-degree result as the classic level-ground benchmark, then adjust geometry or drag in a separate model when those effects matter.

How to enter speed and gravity correctly

Speed is the full launch magnitude in metres per second, not a separate horizontal or vertical component and not a kinetic-energy figure. It must be strictly positive and finite; a zero launch speed produces no trajectory and is rejected with an invalid-input error rather than returning a zero range that could look like a successful solve. Gravity is the constant downward acceleration in metres per second squared.

When you omit it, the capability uses the conventional standard of nine point eight zero six six five metres per second squared, which matches common engineering tables for Earth at sea level. You may override gravity for lunar homework, classroom problems that fix g at ten, or game worlds with custom acceleration, as long as the value stays finite and strictly positive. Keep units consistent throughout: metres and seconds for every quantity.

Mixing kilometres per hour for speed with a g expressed in centimetres will yield a number that is arithmetically correct but physically meaningless for the problem you intended.