Sunday, December 14, 2014

[geqjwbso] Large dense cannonball

Consider the problem of launching a projectile as far as possible.  A projectile encounters air resistance proportional to its cross sectional area.  However, the air resistance affects the projectile as a proportion to its total momentum.  Its total momentum is proportional to its mass, or its volume.

Given the difference between volume (growing by the cube of the linear dimension) and area (by the square), for a given velocity, density, and shape, a larger projectile will travel further.  A denser projectile will travel further.  For a cylindrical projectile that remains facing the correct direction (e.g., stabilizer fins), a longer projectile, having more mass but the same cross sectional area, will travel further.

Motivated by the question, how far can a trebuchet launch something?  By the above analysis, there is no upper bound, so long as one can choose the shape and size of the projectile.

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