Maximum Range of a Projectile Thrown from Constant-Speed Circular Motion

The problem of determining the angle θ at which a point mass launched from ground level with a given speed v₀ will reach a maximum distance is a standard exercise in mechanics. There are many possible ways of solving this problem, leading to the well-known answer of θ = π/4, producing a maximum range of $D max = v 0 2 / g$⁠, with g being the free-fall acceleration. Conceptually and calculationally more difficult problems have been suggested to improve student proficiency in projectile motion, with the most famous example being the Tarzan swing problem. The problem of determining the maximum distance of a point mass thrown from constant-speed circular motion is presented and analyzed in detail in this text. The calculational results confirm several conceptually derived conclusions regarding the initial throw position and provide some details on the angles and the way of throwing (underhand or overhand) that produce the maximum throw distance.