Newton’s Cradle: Colliding football helmets : Physics 101

Parents should absolutely try to improve how to better protect their children from sustaining cerebral concussions. There is not a lot of good basic science out there to address the actual physics of contact, especially for North American football. Imagine a parent going to a sports store asking for a football helmet. What about fit, what about size, what about the liner, what about mouth guards  -do they make any difference ? What about the physics of collision, does the store clerk know what Young’s Modulus means for deflecting collision into the helmet ? Well they should. But not one single source can give good quality information on why only a hard outer shell with a slung or soft liner is the single design feature for all helmets. Think of a falling coconut with its outer soft fibrous shell then the hard shell inside with the inner soft fruit layer, we tend to do exactly the opposite when making helmets. Until this fall, when a patent pending company Guardian Caps began offering an extra outside cover designed for reducing football helmet contact momentum transfer, especially for football practices wherein  the majority of concussion producing impacts occur and probably most of those minor concussions are still unreported.

I will be citing from a pdf  file from a presentation made to the National Football League by Aragon Elastomers a Hanson Group Company, LLC  authored by Chuck Demarest on the Collision Physics for football helmets made December 8, 2010. The Guardian Cap was invented by Lee Hanson of POC Ventures, LLC, Alpharetta, Georgia.

The specific recommendations concern an index termed the Head Impact Criteria (HIC) that currently is not part of the SNELL Foundation that tests helmets in the United States. The presentation topic concerned adding a outer soft ‘cap’ attached to an existing helmet creating a soft-hard-soft interaction different than the classic hard-soft interaction. Here is the executive summary from this report: ” The ‘soft/hard/soft’ helmet construction can deliver better safety than the ‘hard/soft’ construction. An outer ‘soft’ material of the proper density, stiffness and energy absorbing properties reduces the initial severity of the impact. The intermediate hard shell then has lower forces transmitted to it, and in turn conveys lower forces to the interior soft helmet structures and to the head.

Although the soft/hard/soft design reduces the available thickness of the cushioning interior soft structures, the reduction in force and acceleration delivered to the hard layer more than offsets any drawbacks of a thinner interior soft layer. Prototype testing with fully assembled helmets on test fixtures confirms the improved HIC results of the soft/hard/soft design.”

First some basic mathematics about collisions, contact decelerations, rebounds, deflection elasticity and inelastic impact kinetic energy. I hope I didn’t scare you away. Pay attention it’s your child’s brain that I’m talking about providing better, smarter protection.

The cited report uses two examples of two football players colliding helmet to helmet. Assuming a human head weighs approximately 11 pounds, the assumption is both heads with helmets on weigh 16 pounds each. The fastest speed that a footballer will run is about 15 miles per hour which is approximately 22 feet per second which was the closing speed for each player charging into each other on a head to head contact trajectory. The other assumption is that the helmets are capable of deflecting only one inch in this particular collision. (22fps x 22fps = 484 feet squared seconds squared, the deflection calculation is (1 inch x 2)/12 = 0.167 feet)

If you divide 2900 feet per second squared by the constant of gravity, which is  32 feet per second squared each helmeted head will undergo 90 times gravity force of deceleration lasting a period of 7.6 milliseconds from the basic formula of uniform acceleration. That’s a lot of kinetic energy that needs to be shed away from the delicate softness of a child’s brain.

Do you remember that I mentioned in an earlier essay that woodpeckers experience 100 times more deceleration than this as they slam their beaks into a tree looking for ants to eat? These woodpeckers do not suffer concussion effects to their brain which is phenomenal. They don’t have a soft/hard/soft helmet to deflect this huge brain jarring force. But let’s get back to the colliding helmets. In the real world this deceleration is not a consistent nor constant calculation but an estimate of an average which can actually be a higher deceleration collision.

The force in pounds is derived from the formula Force =m* a. (mass multiplied by acceleration) In this case the weight in pounds is divided by 32 resulting in ‘slugs’ so 16 pounds total helmet+head divided by 32 times the acceleration of 2900 feet per second squared is 1,450 pounds applied for 7.6  milliseconds, assuming uniform deceleration to make things simpler.
Also involved with these forces and accelerations are Energy and Momentum. Recall the Rule of the Conservation of Momentum, is always respected:


The Law of Conservation of Momentum states that the total linear momentum (p) of a closed system is constant. That means that the momentum of the Newton Cradle steel balls on impact equals the momentum of the second group of steel balls after impact: (I’m using the swinging Newton’s Cradle steel balls to illustrate momentum transfer during the colliding heads-this way I think you can visualize the collision events more readily)

p = mv = MV

Shows that the number of balls moved are the same

Solve for v and square both sides of the equation:

v = MV/m

v2 = M2V2/m2

Substitute v2 in the energy equation mv2/2:

mv2/2 = mM2V2/2m2 = M2V2/2m


mv2/2 = MV2/2

M2V2/2m = MV2/2


M/m = 1 or M = m

This means the mass of the balls leaving equals the incoming mass. Since the balls are of equal mass, that means the same number of balls leave the series as those which impacted the group of balls.

Also, from the equation: V = v

( sourced from

Remember that in the ‘inelastic’ collision the kinetic energy is dissipated as heat. Also remember that a human head is also hard/soft so that even if a deforming substance on the outside of the head could transform completely into heat the soft brain still decelerates into the hard boney skull. The real observation of a deforming substance used to interact with the collision mass and velocity is this: upon initial contact a high rate of deceleration will occur but this rate of deceleration decays as the substance deforms, it’s the nature of the deforming substance as an ‘inelastic collision’ that determines the decay time factor of the absorbent material used for the deforming interactions occurring during the collision.

During an ‘elastic’ collision it’s like the classic steel balls suspended touching each other apparatus, called Newton’s Cradle.  If the first ball is raised then released all the energy transfers through each ball swinging into each immediate neighbor’s steel ball. If the collision is completely elastic without any energy being absorbed the velocities of the helmets instead of steel balls are exchanged. The elastic collision is like two rams butting heads, they jar backward, stunned, as the impact speed turns into the rebound velocity, with virtually no kinetic energy transformed into heat except if the horns engage and stick their friction would become heat. This situation is the worst for both rams and for two colliding helmeted football players since the transfer from impact into rebound force lasts twice as long. During this delay the brain sloshes into cranial bone deforming/shearing in the coup/contre-coup brain rebound.

What happens if one helmet/head is at rest while there is an impact from another helmet/head at the same velocity of 22 feet per second? Momentum is contained within only the moving helmet but now must be divided between the two helmet/heads after collision. If the collision is completely elastic the stationary helmet is accelerated at 7.6 milliseconds to 22 feet per second as the velocity transfers from one helmet/head into the other helmet/head, assuming no energy is absorbed. The moving helmet/head stops completely after the collision. If the moving helmet/head collision is completely inelastic then both helmet heads, in order to satisfy the Conservation of Momentum, would have half the speed of the impacting helmet/head or 11 feet per second. The limit of the dissipation of kinetic energy as heat is now half the energy of this particular collision. Kinetic energy is proportional to velocity squared, thus each helmet/head with half the velocity during the rebound is half of the original kinetic energy of motion of the two helmet/heads plus the remaining other half is dissipated as heat. Notice that the velocity speed of the accelerated stationary helmet/head is half compared to the elastic collision example.

What about an ‘inelastic collision’ ? Suppose the total mass of the helmet and head are made with elastic behaving clay, like PlayDoh. What happens during a head-on collision between two clay covered helmet/heads? Both helmet forms would deform stopping at the point of collision. The clay would be hotter since the kinetic energy of motion would have transformed into heat. However for helmet makers this becomes the crucial aspect of the deceleration. As the clay helmet/heads impact, the clay material will have a very high rate of deceleration at contact that rapidly decays with time as the deceleration slows. Trying to maximize this absorbing deceleration capacity is the holy grail for helmet researchers trying to find the magic combination of absorbent material to dilute out the deceleration rate from rapid/slow- stretching the deceleration into slow/slow.

The key observation between two helmet/heads interacting at elastic and inelastic collisions is to minimize the acceleration of the brain by minimizing the transfer of momentum between the two colliding helmet/heads. If a helmet/head has an inelastic energy absorbing layer as a added soft layer surrounding a hard shell of a helmet, this extra soft layer will accomplish just such a reduced momentum transfer between the two helmet/heads. Both helmet/heads will have diluted forces transferring into the elastic brain. This is how the Guardian Cap works.

The holy grail for helmet makers is establishing the best absorbing material to cause diluted acceleration at the transfer contact between helmet/head collisions depends on the absorbing materials characteristics to deflect and deform within a stiffness performance during the contact. The reaction is a complex mix of inelastic within elastic properties which ultimately alter the rate of deflection combined with the amount of deflection. The precise term for this stiffness is termed the Complex Modulus.  This property of stiffness is a function of the absorbing materials, combined with the rate of deflection and the amount of compression expressed as a percent of shape change for the  absorption material’s performance during the collision. Here’s their mathematical relationship:

Complex Modulus = f (Rate of Deflection, Young’s Modulus, % compression)  Young’s modulus is the ratio of stress, which has units of pressure- to strain which is dimensionless; therefore, Young’s modulus has units of pressure.

Just like ancient Carthaginian soldiers holding their deflecting shields known as parmas and scutums capable of deflecting mortal blows to soldiers during hand to hand combat, the same laws of physics apply to reducing blows to the head during football combat-like collisions between players helmet/heads. The Guardian Cap is much more than a mere cover, it is performing like an shield over the helmet/head by actively reducing the contact deceleration energy into the head. The Guardian soft shield is a significant improvement toward reducing forces into the head.

The Guardian Cap has been invented to reduce helmet collision momentum transfer.

photo by Jaren Haser/ Sports Editor Lake Country Leader

So if you’re a parent worried about the real details for helmets
you’ll have to ask for the best Young’s Modulus protection covering for helmets to reduce the momentum transfer. It’s all about extra protection and it’s all about the physics of collision impact toward hopefully reducing cerebral concussions, especially for younger participants.

Guardian Caps Helmet Protection Cover

About cerebrovortex

Montreal Grandmother, Agnes Kent was saved by Raul Wallenberg from certain death, when he provided papers for her and her Mom to escape away from the Nazis. Today when asked what that escape meant, she replied,"Remind people, that while statesmen and whole countries remained silent and did nothing, a single individual chose to act, with ramifications that proved enormous. Similar choices confront us today. Write that simple truth she said, it can never be repeated often enough because the world keeps forgetting it."
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4 Responses to Newton’s Cradle: Colliding football helmets : Physics 101

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