no minimum.....if the bullet punches deep enough to disrupt the plumbing the critter dies......the whole minimum energy thing becomes real stupid once you realize ppl kill the damn things with arrows......[bleep] up the plumbing and you have a dead elk.....
You can't compare a bullet to an arrow. The mode of killing is totally different. An arrow cuts as it goes through and penetration is much more unless it hits a big bone.
An arrow lacks the energy of a bullet, but it has much more momentum because of it's much greater weight.
