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$KE = \frac{1}{2}mv^2$ meaning the KE lost on impact would increase quadratically with speed, but f = ma would mean that force the wall applies to the car increases linearly with speed. So which really determines the damage to the car itself?

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marked as duplicate by JMac, stafusa, Kyle Kanos, Bill N, John Rennie energy Jan 10 '18 at 8:06

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$F=ma$ does not imply that the force increases linearly with speed, because the time it takes the collision to happen is not constant. If the distance that the car crumples is constant, then the average force during the crash is quadratic in the speed. If the time it takes were constant, it would be linear. The actual situation is likely somewhere in between. "Damage" does not really have a quantitative definition, so it is impossible to say that it is exactly proportional to energy. But it is safe to say that it goes more like energy than like momentum.

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Energy is never lost because of the law of conservation of energy. Energy just changes from one form to another. So, I guess that you could say that the force "determines" the damage; since the wall cannot be deformed, the energy would go into processes like "damaging" the car and the sound of the impact.

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The force varies from Point to Point of the car and moreover, the total force is not a reasonable indicator about how much damage the car takes. Much better is the value of the stress $\sigma$ which is defined as force per area. Thus, a car with a larger area will have less stress (at equal force) than a car with small area.

There are material Parameters like the critical stress, e.g. yield stress that indicates the minimal stress where plastic deformations occur.

Energy in a plastic Deformation is also a plausible quantity. It is defined as

$e = \int_0^{\epsilon} \sigma d \epsilon$ (energy per volume)

with the strain variable $\epsilon$. It measures the energy that is required to deform the car by the strain $\epsilon$; however,, you must subtract the elastic (i.e. reversible) energy to obtain plastic energy. If you draw load cycles in a stress-strain diagram, plastic energy is the area enclosed bythe lines that depict the load cycle. Thus, you have

$e_{plast} = \oint \sigma d \epsilon$

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the dynamics of how a car crumples up during a head-on collision with an immovable object are extremely messy, but to answer the question does not require detailed knowledge of that process. if you assume that the wall is immovable, then by the conservation of energy we have

(kinetic energy contained in moving car before collision) = (total work performed in deforming the car's components)

so the damage to the car is determined by the car's kinetic energy.

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f = ma does not mean that force is linear with speed, and if force were to increase linearly with speed, then the distance over which it acts has to also increase linearly with speed. It is reasonable to expect that the damage will therefore scale more than linearly with speed. The exact amount of damage depends on the structure of the car.

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