I have spent hours days and weeks on reading about what physical laws that determines the penetration of an arrow. The more extensive essay on the subject concluded that only Momentum was the important factor when determining if an arrow will penetrate what it hits or not. Then other smaller essays conclude that only Kinetic Energy determines the success of arrow penetration. Obviously we are missing something here.
I then thought about it and quite simply understood that penetration is two-fold. First at the very moment the arrow-head connects with the surface of the target. The most important physical relationship here, as in all other similar cases, is the pressure. pressure is the relationship between applied force and the contact area of the applied force. The area is derived from the arrow-head’s sharpness (the sharper the smaller area is in contact) . The force can be verified either from the momentum or from the kinetic energy, it matters little for the resulting force will be the same. Then it is the ratio of the target’s Hardness and the arrow head’s hardness that determine how much pressure is needed.
Now, once the arrow has penetrated some other phenomena appears. As the arrow head now is not only in contact with the target with its tip, it is also in direct contact with the sides of the arrow. Imagine you pushing a finger between two other fingers. It is the sides of the finger that is in contact with the fingers, not the tip. When this happens Friction will happen and act as an opposing force. From this I conducted an experiment.
I took a shelf and a book, leaning the shelf in an angle and putting the book to let it slide down. I measured the distance the book travelled and I measured the time it took for the book to travel the noted distance. I then increased the leaning angle of the shelf and found, naturally, that the book travelled the same distance much faster.
From this I could then calculate the ratio between the speed och the book and how long it would travel in the same time. This gave me the exact relationship of Distance is equivalent to Velocity to the power of 2. In other words. If you increase the velocity of the arrow to twice its speed, it will take it 4 times longer to be stopped through friction compared to the lower speed.
This inevitably means that Kinetic Energy decides how much an arrow will penetrate. However, the mass of the Arrow still has a very important role in this and I have yet to figure out how important it is and the relationship it has to the penetration.
If you are interested in the experimental data and the calculations, please let me know.