range

The range of a projectile is the horizontal distance it travels between launch and return to its starting height. For a shot fired from level ground at speed v and angle θ in vacuum, the range has a compact closed form: R = v²·sin(2θ)/g. The formula shows two things at once. First, range scales as the square of the launch speed, so doubling the muzzle velocity quadruples how far the shot goes. Second, the angular dependence sin(2θ) is symmetric about 45°, so the maximum reachable distance is R_max = v²/g, achieved at θ = 45°, and any pair of complementary angles (θ and 90° − θ) gives exactly the same range.

In practice, projectiles almost never launch and land at the same height, and the air is never absent. If the launch height is above the landing height — a shot-put released from shoulder height, an arrow fired from a hill, a ball thrown from a cliff — the optimal angle drops below 45°. If drag matters — as it does for any fast projectile in air — the optimal angle drops further still, typically into the 30°–40° range for sports balls. Artillery fire tables from the seventeenth century onward tabulated ranges for a grid of angles and charge weights, because no simple formula suffices once the physics is realistic.

Evangelista Torricelli compiled the first systematic projectile range tables in Opera Geometrica (1644), applying Galileo's parabolic theory to artillery practice. He also discovered that for a fixed launch speed, the family of trajectories at every possible angle is bounded from above by another parabola — the safety parabola — outside which no shot can reach.