The Ballistic Coefficient of Pumpkins

I catapulted a pumpkin into the air,
It fell to earth, I know not where.1

― Apologies to Longfellow2

3

A trebuchet is a type of catapult that converts the potential energy of a counterweight into the kinetic energy of a projectile. The simplest version operates like a see-saw, with the counterweight suspended from a hanger attached to the short arm and the projectile held in a sling attached to the throwing arm. When the short arm is raised and then released, the throwing arm rotates faster because it is longer and the sling rotates even faster as it whips around the end of the throwing arm.4

At their inception during the Middle Ages, trebuchets were used to hurl boulders at castle walls in an attempt, usually successful, to batter them down. Modern trebuchet designers, lacking castles to besiege and fair maidens to rescue, must content themselves with hurling cooking pumpkins for distance. Competition, however, is fierce, and the winner of the contest can expect any fair maidens present at the pumpkin festival to hurl themselves at him. Thus, it behooves us to put as much study into the ballistics of pumpkins as old-time mathematicians put into the study of cannonballs.5

From wind tunnel tests we know that the coefficient of drag, CD, of a pumpkin-sized cannonball is 0.2. This is not bad, as smaller spheres, such as musket shot, have a CD of 0.45 due to the fact that their Reynolds number is below the critical point where the air flow in the boundary layer changes from a smooth motion to a turbulent one. Somewhat counter-intuitively, a turbulent boundary layer carries more energy and is thus able to remain in contact with the surface a bit longer. This narrows the wake and increases the pressure behind the sphere. The reduction in form resistance more than compensates for the increase in viscous resistance, so there is a net reduction in drag.6

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