The exact same thing would happen even if you did shoot the two arrows from the same bow. The lighter arrow would lose a significantly higher percentage if it's speed and energy than the heavy one would.

The lighter arrow is faster. Increase the speed, you increase wind resistance. Air resistance increases in proportion to the square of velocity. The heavy arrow is slower and has less wind resistance. It doesn't have to spend as much of it's energy cutting through the air as the light arrow. PLUS it's additional mass allows it to use even less of it's energy to cut through the air resistance.

As I've often pointed out, if light arrows can't even penetrate AIR as well as heavier ones, how could anyone think they penetrate as well on game? That's another side of the same issue.

Here's something to peek at for a bit of background on archery ballistics.
http://home.att.net/~sajackson/archery_physics.html