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wrote:
Ted Edwards wrote:

Bob Swinney wrote:
Fire a cannon ball exactly horizontal (level) with the earth. At the exact
instant the cannon ball leaves the muzzle, drop another cannon ball from the
same height. The 2 cannon balls will reach the earth at the same time.

On an airless planet but not on Earth. Think Areodynamic drag.

Even on earth.

Nope.

Basic physics has us break things up into components.

This is frequently the case and examples in high school Physics have
carefully chosen conditions so this will be ok. In order to do so, we
apply the principle of superposition which requires that the total
effect of a set of forces is the sum of the effects of the individual
forces. This requires linearity. Drag is proportional to velocity
squared.

Set up the differential equations of motion for a projectile fired at
some angle to the horizontal in a vertical gravitational field and
experiencing a retarding force alligned with but in opposite direction
to the motion vector that is proportional to the magnitude of the
velocity squared. It's a neat problem and fun to work. (I first did it
studying arrow flight.) You will discover two things: 1) The resulting
set of differential equations has no closed form solution (elliptic
integrals) and must therefore be solved numerically. (Runge-Kutta is a
good method if you wish to try it.) 2) Attempting to separate the
forces into vertical and horrizontal components gives an approximation
of variable quality but is, in fact, wrong. Speaking vaguely, since the
aerodynamic forces depend on the square of velocity, a change in
horizontal velocity affects vertical as well as horizontal force and
vice versa.

Ted