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If a rock is thrown at someone's head, or a car drives against a wall, or a meteor hits a planet, is the physical damage done to the object proportional to the projectile's momentum, or to its kinetic energy? (assume that the target object is at rest in the reference frame).

I'm sure that it depends on all kinds of factors, but roughly, does it depend linearly on the projectile's speed (as momentum does), or quadratically (as kinetic energy does).

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    $\begingroup$ Neither. It depends on a lot of things. Shape of object. Orientation of object. Relative velocity. ... $\endgroup$ – Brick Aug 5 at 18:11
  • $\begingroup$ @Brick, edited. $\endgroup$ – user56834 Aug 5 at 18:21
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    $\begingroup$ Still neither... $\endgroup$ – Brick Aug 5 at 18:21
  • $\begingroup$ physics.stackexchange.com/q/141779 $\endgroup$ – GenlyAi Aug 6 at 0:10
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It is proportional to the energy (but also depends on the nature of the collision &c &c)

Here's an example of why. Let's consider the kind of case proper physicists spend all their time thinking about: giant asteroid impacts (even now, teams of mad physicists, all played by Peter Sellers, are working out how they can steer some asteroid to collide with the Earth thus rendering all non-mad-physicists extinct).

The thing that makes these really bad is heat: when some large meteorite collides with the earth it's not the case that the Earth gets knocked out of its orbit and falls into the Sun, or gets thrown into outer space, or that the impact is so violent that everyone falls over or something (I have no idea what the seismic effect of the K-T event was, but I do know that it wasn't what did for the dinosaurs). What causes the problem is the abrupt release of a very large amount of heat: about $10^{23}\,\mathrm{J}$ of it. This caused some pretty spectacular events. There is dispute as to whether the impact caused the entire terrestrial biosphere to catch fire, but, well, there is dispute about it: it may have done that.

But you can't turn momentum into heat: momentum is a vector quantity and it's conserved: all you can turn momentum into is momentum. What you can turn into heat is kinetic energy. So it was not the momentum of the Chicxulub impact which did for, well, pretty much everything, it was the kinetic energy.

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"Physical damage" would be more linked to the solid mechanics of the object. Depending on the material, the object would have varying "physical damage."

From physical damage, an object's material changes. The momentum and the kinetic energy of an object changes when physical damage is done, but this is more related to the acceleration and force of the object during the collision.

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Fools rush in where angels fear to tread. An exceptionally crude model of a body to be deformed would be a compressible spring (of spring constant k) with one end fixed to an immoveable object. When a body with mass m and speed v makes a head-on collision with the free end of the spring the spring compresses by x in which $$\tfrac12 kx^2=\tfrac12 m v^2\ \ \ \ \text{so}\ \ \ \ x\propto \sqrt{KE}.$$ We could, perhaps, take the compression x as a measure of the damage, so the damage is proportional to the square root of the colliding body's KE..

Note that this model gives no relationship between damage and momentum, as the colliding body's mass and speed are involved differently.

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