There is absolutely nothing contentious about what I have written. If you think so you need to go back to look at those BBC bitsize videos I recommended. You do know that momentum of an object is it’s mass x velocity? Do you agree that’s the definition? Do you agree that an object can come to rest after an accident? Do you therefore agree that it’s momentum will be less?
The speed that the brain slows relative to the helmet is irrelevant here. But it also does slow down don’t you agree? Or does it keep it’s momentum which I recall was your origional argument that the brain had all the momentum as it was transfered to it.
You get a far clearer picture thinking about it all in terms of kinetic energy.
KE is energy possesed by a body because it is moving and is equal to half of the mass multiplied by square of velocity.
So, as I’m trundling along on my bike, my brain has some KE. If my head comes to a halt, the kinetic energy possessed by my brain has to fall from whatever it was to zero, the kinetic energy has to be dissipated.
If I come to a halt gently, using the brakes, the KE from my brain (and the rest of me) is dissipated harmlessly as heat at the brake pads. If I come to halt suddenly, the KE possessed by my brain is dissipated suddenly. The most extreme deceleration probably occurs if I come to a halt by bashing my head on something, and in that case pretty much the only thing reducing the rate of deceleration of my brain is the fluid inside my skull. All of the KE possessed by my brain is dissipated near-instantneously as heat and tissue damage as my brain contacts the inside of my skull - not good!
Adding a cycle helmet of the conventional kind doesn’t greatly influence the rate of deceleration if my brain, because they contain no “crumple zone” to reduce deceleration and dissipate energy, although see below regarding helmet breakage, and some designs of helmet can add extra trouble by causing rapid twisting of the head under certain circumstances too, thereby causing tearing of the lining of the skull.
So, don’t bother with a helmet?
Well, that wouldn’t be my answer, because a helmet does offer some protection against cuts, penetrations etc, and it can dissipate energy by breaking apart. This last point isn’t trivial, because it takes a fair bit of energy to split or break a cycle helmet, and energy dissipated that way is energy not harming the wearer. It probably saves skulls from getting broken.
In short, I believe a helmet can protect your head from some forms of injury, but isn’t much help in regards concussion. As a BTW, you don’t even need to hit your head to suffer concussion - provided the deceleration of the head is too quick for energy to be dissipated harmlessly, it will occur.
Motorcycle and other full-face helmets often include some degree of crumple zone padding, but still not much, so again might not be proof against concession in a very rapid deceleration, but they are designed to prevent the wearer getting their jaw smashed, and some also act to reduce neck crushing and twisting.
As a footnote, worth noting that the velocity component in the KE equation is squared, which means that speed has a huge influence on what injuries occur in a collision. My surmise/guess/hypothesis is that human beings have evolved anatomy that copes very well indeed with the energies involved in falling over when walking, and pretty well with the energies involved in a wide range of running accidents, but that our anatomy is severely challenged by the energies involved in, for instance, cycling accidents at greater than running speed, because the extra weight and complexity of tissues to cope with that would impose an evolutionary penalty and get selected-out if/when they arose. I often dont wear a helmet when riding my ponerous old shopping bike gently on paths away from roads, because the speeds and hence KEs involved are comparable with those of running not very fast.