[podperson]

It's an interesting post, but perhaps a simpler way of looking at the whole thing is you should be using average velocity over the time elapsed rather than the just-calculated new velocity.

This becomes:

velocity_new = velocity_old + acceleration * time_elapsed

position_new = position_old + (velocity_old + velocity_new) * 0.5 * time_elapsed

It makes immediate, intuitive sense (at least it does to me) and doesn't require even thinking about differential equations (at least for constant forces).