# Need table plotting decelerating velocity & appx position in milliseconds

1. Apr 21, 2008

### S_I

An object weighing 34.518379357 kg, free-falling at 4.2(?) m/s, travels 0.889m, before it begins decelerating when it makes contact directly on the top of a person's head.

Within what I would guess would be perhaps maybe 2(?) milliseconds after deceleration commences, the first intervertebral disc between C-2 and C-3 vertebrae begins loading up.

It's tensile strength is instantly overwhelmed as the impact force of 11986.30338302126 N forces the inner nucleus pulposis outward radially, generating fissure pathways through the 13 concentric rings of lamellae comprising the progressively more fibrous Annulus Fibrosis, until the inner disc material breaches its outer wall, and also breaches the immediately adjacent central posterior ligamentous capsule, creating what is known as a transligamentous disc extrusion pressing into the spinal cord. (Ouch!)

This process repeats as the impact force continues downward until the energy is finally absorbed 9 intervertebral discs later. (ouch x10!)

The estimated cumulative height of the affected intervertebral discs is 24.5 mm.

Treating the 10 discs as though their heights were the same (even though they are not), how would one APPROXIMATELY plot their decelerating velocity and position in milliseconds?

Since I am an absolute beginner in physics, I cannot even begin to throw out an idea of where to start. Can anyone help?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Apr 22, 2008

### astrorob

This seems quite an involved problem which needs to simplified before proceeding..

Can you treat the ten discs as being one body? That is to say, when the impact force acts on the discs do they all experience:

i) The same force as one body or are they better approximated as ii) individual discs connected by "springs" for example?

3. Apr 22, 2008

### S_I

You are correct. Treating the discs as one body would not yield very good approximations.

I did not make that very clear. When referring to treating the 10 discs "as though their heights were the same" I'm pointing to the fact that each consecutively descending disc is designed to support a slightly greater load bearing capacity. As they reach the bottom of the cervical region they are taller than at the top. And once below the cervical region the thoracic discs begin becoming noticeably heftier.

So to carry the "spring" metaphor forward, it would be like placing increasing heavier gauge springs (i.e. subtly wider diameter coils with say one additional turn of steel in their heights) on top of each other between rigid concave ended cylinders (vertebral bodies) of similarly increasing size.

When saying "as though their heights were the same", I'm assuming their average would be reflected in their combined height and an average of the two in the center. Clearly the plot would be different for individual "springs" of equal dimensions and gauge than progressively increasing ones.

I just don't have access to scientific data on the approximate height of cervical and thoracic disc height averages and have not found any data in google searches which is sufficient.

That is why I say to treat them as the same individual heights. I will hopefully get that other data at a later time to adjust the table plots accordingly.

4. Apr 22, 2008

### astrorob

OK, I see.

I propose the following model:

Consider a series of masses interconnected by springs. The masses all have a maximum load force which is equivalent to saying that the mass doesn't move unless this force is reached.

The first mass, as you put it, is instantly overwhelmed by the impact force. This force is translated into translational motion of the first mass by Newton's first law (N2L);

$$F = ma$$​

The mass now starts to compress the first spring. With this compression comes a restoring force which acts to oppose it. This is given by Hooke's law;

$$F = -kx$$​

Where k is the "spring constant", arbitrary at this point, and x is the displacement of the spring from its equilibrium position. As you can see, increasing the displacement increases the restoring force, so as the spring becomes more compressed it has a greater restoring force.

Until the maximum load force of the second mass is reached (that is to say until the first spring is compressed sufficiently so that its restoring force is greater than it) then the second mass will act effectively as a wall. After the load force is reached, this second mass will begin to move and the system proceeds as such for all the masses.

With this model currently, there will be "oscillation" of the discs around a fixed point since there is no loss in energy from the system. (Think of masses interconnected by springs dangling from a clamp. Set this in motion and it would continue to oscillate forever but it doesn't because of air resistance. You'd need an equivalent dampening force to stop this from happening).

Do you think this could accurately describe the system?

Edit:

This may aid visualisation:

http://www.myphysicslab.com/spring1.html

Set the 'damping' to something like 1.

Last edited: Apr 22, 2008
5. Apr 22, 2008

### S_I

Thank you Astrorob,

I do believe you have either entirely solved this or I won't know if I have any future questions until I have thoroughly absorbed the implications of your proposal - something which will keep me busy for quite a while.

Being the quintessential newbie to physics it took me quite a while to even gather the initial data and learn how to present some rough approximation of a marginally coherent question.

So from what I can tell at this point it's all about getting my head wrapped around some very basic things which I have never studied before. Asking any questions at this point would only reveal my ignorance far more than I already have. By the time I have any clue as to what if any additional questions may come, this thread will likely be closed.

Ultimately, after grasping the essence of the math and finding the specifically applicable data set, I'm going to have to communicate this to a neurosurgeon in terms which hopefully won't go too far over his head, as it presently does over mine.

Thanks for pointing me in the right direction. It looks like the rest is just me getting up to speed...

Regards

6. Apr 23, 2008

### astrorob

There's nothing ignorant about not knowing a discipline you've never studied! The model I've actually proposed there may use some basic concepts but the motion it leads to would actually be quite complex (I imagine).

Glad to have been of help,

Rob.

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