Ted Edwards wrote:
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

Particularly in the extreme case of a bullet fired a few feet above the
ground horizontally.