There’s no short answer to this. To analyze ‘damage’ we normally turn to simulations or practical testing.
Simulations usually involve creating a program that accurately simulates every element (for instance every beam and panel in the front of a car). Then simulating the movement and deformation of each piece using the very simple equations of force most people learn in school.
Often, from simulations and real testing, we can draw up heuristics – approximations using math that are based on the test results. In some circumstances, we can draw conclusions like “faster cars take more damage” or “more rigid cars injure passengers more” or even “larger bullets injure people more”. Sometimes we can even get equations, like “the depth a projectile penetrates into a material scales linearly with its density and length” but even these are limited to specific situations.
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