Range of Projectile Calculator

The Range of Projectile Calculator computes the horizontal distance a projectile travels when launched from and landing at the same height on level ground. This fundamental kinematics problem assumes no air resistance and constant gravitational acceleration, providing the idealized baseline for ballistic calculations.

The horizontal range is given by: $$R = \frac{v_0^2 \sin(2\theta)}{g}$$ where $$v_0$$ is the initial launch speed, $$\theta$$ is the launch angle measured from the horizontal, and $$g$$ is gravitational acceleration. The $$\sin(2\theta)$$ factor reveals the elegant angle-range relationship at the heart of projectile motion.

A remarkable property of projectile range is that complementary angles produce identical ranges. An object launched at 30° travels the same horizontal distance as one launched at 60° with the same speed, because $$\sin(2 \times 30°) = \sin(2 \times 60°) = \sin(60°) = \sin(120°)$$. This symmetry breaks down when air resistance or launch height differences are included.

Maximum range occurs at exactly 45°, where $$\sin(2\theta) = 1$$, giving $$R_{\max} = v_0^2 / g$$. This is a direct consequence of the equal partition of velocity between horizontal and vertical components. In practice, optimal launch angles are lower than 45° due to air resistance—competitive shot putters release at about 37°, and long jumpers take off at approximately 20°.

The range formula implicitly assumes the projectile lands at the same elevation it was launched from. When launch and landing heights differ, the range equation becomes more complex and the optimal angle shifts. Launching from elevation (like artillery on a hill) increases range, and the optimal angle becomes less than 45°. Conversely, launching uphill steepens the optimal angle.

Projectile range analysis has profound applications in sports science (optimizing throw distances), military ballistics, rocket trajectory planning, forensic analysis of falling objects, and even geological studies of volcanic ejecta. The calculator also shows the time of flight and maximum height to provide a complete picture of the trajectory.