The following post was begun on my parkour blog but left incomplete and never published. I've copied it over and added a little bit to finish it off. I've ended it rather more sharply than intended because I don't have in mind the original thoughts and have instead used my current thinking on rolls, combined with my physics knowledge.
The old bit:
Rolling is one of the techniques that are most commonly seen by beginners in their introduction to parkour. The skill is not particularly difficult and it is essential to learn early on for safer landings. This article will focus on the main purpose of the roll in parkour, one of the major misconceptions of how a roll decreases impact on the body, and then finish with a description of the technique for beginners to learn. More experienced practitioners may like to use this technique for developing rolls on their weak side.
What is the purpose of a roll in parkour?
While there are different uses for the roll in parkour, e.g. dive rolls, fast direction changes, etc. we are considering the roll as used during the landing of a jump. There are two main uses within this category: to help reduce horizontal momentum for a high speed landing where it is impossible to remain standing, and to reduce the forces experienced by the body when jumping/dropping from a height. Our primary consideration is the latter - reducing forces on the body when landing from a height.
Misconceptions
One obvious misconception is that a roll will save you no matter how high you drop from. More experienced traceurs often joke about being able to jump off the tenth story of a building and rolling in the landing to save themselves. While it seems pretty obvious that there are limits to the ability of a roll to reduce impact on the body, many people don't understand exactly why this is. The answer lies in the way a roll reduces impact forces on the body and what muscles in the body are engaged (and to what extent) during a roll compared to landing without a roll.
The new bit:
The other main misconception is that rolls transfer vertical momentum into horizontal momentum, and that this protects the jumper from injury. Now, it is true that injury can be reduced, but the reasons are not to do with a vertical to horizontal transition. Any momentum in a vertical direction must be removed by a force applied in the opposite direction. And any momentum gained in the horizontal direction must be added by a force applied in that direction. The vertical and horizontal are independent of each other in such analyses.
So the body experiences reaction forces in the landing which remove the vertical momentum. The longer the force is applied, the greater the change in momentum. Or for the same amount of momentum, the longer the force is applied, the lower the force needs to be to remove the vertical momentum. The main reaction force is upwards from the ground and this transfers through the body's natural suspension system of joints, ligaments, tendons, and muscles to halt the descent. From the physics we know that the greater the instantaneous force, the higher the chance of breakage.
Here's the final conclusion of all that science:
A parkour roll used in a vertical landing works because it reduces the maximum instantaneous force used to remove the vertical momentum from the drop. The force experienced by the body is spread out over a greater time during the landing. A roll landing allows the lander to take more time in the landing by crumpling, much the same way that a car bumper is designed to crumple to absorb momentum.
Another way to look at it is with an egg falling onto a brick compared to an egg falling onto a foam mattress. The brick tries to stop the egg instantly because it has no give. The mattress slows the egg more gradually and lowers the maximum force on the egg.
So why bother rolling? Why not just crouch on the landing? Well, for at least three reasons. One is that you can't crouch to bring your center of gravity as low as you can get with a roll, this means a roll is better at increasing the amount of time taken to land. Two is that very often there is some horizontal momentum from the jump which means a roll is a more stable technique for landing. Three is it is possible that more muscles can be recruited to spread the force out between while rolling because a greater body tension is required for the execution of the roll.
This information can inform our rolling technique. Instead of just copying a video, you now know what to do to have the most effective roll. You need to make your landing last as long (and as smooth) as possible to ensure the lowest possible force on your body. This means resisting the landing from the highest point all the way down to the lowest crouch possible before rolling to continue the descent a fraction longer.
It is worth mentioning that the roll is also just smoother and allows you to regain your feet easily. Paratroopers in past wars have used a crumple method of landing that didn't rely on getting to their feet again. They would crumple into a sideways kind of heap like the video below (about 1 minute into the clip):