I presume you mean in "height measurement is how it is related to the takeoff area" that the change in height is related to the original height of the center of mass with respect to some arbitrary point later where we can again determine the height of the center of mass.  If that presumption of mine is correct, then here's how it boils down:

The horizontal and vertical components of a body in motion aren't related beyond the fact that they happen to occur at the same time.  So, for the purposes of determining the change in height, it's completely immaterial what the horizontal component is doing.  In other words, a car moving the speed of sound in the x direction will drop in the y direction (on a Cartesian coordinate system) in the same amount of time that a car stationary in the x direction will drop in the y direction.

Whatever the particulars of the any case one might examine, what happens before we can determine the height of the center of mass, and what happens after we can determine the height of the center mass (provided that the first and second determinations aren't equal) is completely irrelevant.  So, the car could hit a tree, zip up, do a barrel roll, fall down, hit a ramp and do an aileron roll and then go marching down a parade route and none of that data will be at all relevant to determining the second position of the center of mass we need to find.

Now, of course in collision dynamics, we related horizontal and vertical motion together in our equations.  Now, how does this work considering that the two components are independent of one another? The reasoning is more obvious to some than others, but it's essentially because the acceleration of gravity will operate on the vertical component irrespective of what the horizontal component is doing we are able to learn more about the horizontal motion because it isn't effected by gravity.  The horizontal motion is a product of the "speed" of the car.

From this, we can see that the relevant bits of information are the height at which the center of mass of the car was when it went airborne, the angle of departure from earth it took, and the height of the car at some arbitrary (albeit first) post take-off position.  What happens after we find that arbitrary position is immaterial.

KEN wrote:

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So whenever I take my measurements from takeoff to first touch, for the sake of the airborn equation, what happens after the first touch has no bearing on the outcome.

For determining the speed at which the vehicle first went airborne, what happens next has no effect.  I took that to be the question you're asking.  Obviously, if the occupants are very concerned what happens on the next hit, and the next.

 

Therefore if a vehicle runs off of the road and goes airborn and strikes an object, I must determine if the height at which that object was struck was higher or lower than the original center of mass take off point and how many feet difference it is. Correct?

Not quite. What is of importance is not the height of what is being struck.  What we are concerned with is the motion of the center of mass of the vehicle.  From that it follows that what you must determine is whether the center of mass of the vehicle is higher or lower than it was when it left the ground.

To make it a little more clear, consider that you have a car in typical Hollywood fashion coming towards the ground at about a 45º angle, that is to say from the hood to the trunk, the vehicle's geometric orientation is 45º.  So, we have that in mind, and now consider that the vehicle is falling towards the earth, but also moving towards a flat wall of some kind.  Now, the car is going to hit this wall still with its geometry canted at 45º (for the sake of example, clearly this isn't the case once the collision happens).  Now, ignoring that the car is going to pitch longitudinally towards the wall the hood just struck (thus leaving a nice imprint of the roof of the car on the wall), the task is twofold:  1.) determine what part of the car first hit the wall, and 2.) given the geometry deduced from 1.) before, where was the center of mass located?

Now, from this you can readily see (if I've been sufficiently plain), that the point at which the vehicle has struck an object doesn't imply that we can stop there and take our measurements.  Finding the first point of contact is simply the starting point to determining whereat we should likely find the center of mass (bearing in mind that the center of mass might not even be within the vehicle anymore, which is a product of rotational forces).

 

To recapitulate (TL;DR):  we're interested in the motion of the center of mass, not the periphery of the vehicle.  So, from the moment of takeoff, you must determine whereat the center of mass is, and at the moment of first impact, you must determine whereat the center of mass is.  It's the motion of center of mass to center of mass (so long as one doesn't vertically equal the other) which is the information we require.

 

I've tried to edit out all of this empty space, but without any success.  So, enjoy the blank half page of nothingness.

-- Edited by ashman165 on Wednesday 10th of March 2010 03:15:03 AM

-- Edited by ashman165 on Wednesday 10th of March 2010 03:16:00 AM