The Collision of Rocks - Momentum Impact

[bluecheez]

(This is not a homework problem, but is just something I'm curious about.)

If two rocks with different momentum collide, how much "damage" will the rocks receive as a function of their momentums? For example, could I figure out a momentum (in the elastic or inelastic case) that would cause one of the two rocks to shatter without causing the second rock to shatter?

 

[BOYLANATOR]

Whether or not the rocks will be damaged will depend on a lot more than just their momenta. It will depend on their structure and shape, the point and angle of impact, impurities in the structure of the rock, small existing cracks in the rocks etc.

This is not something you can derive from basic physics equations or momenta alone.

 

[bluecheez]

Well, then why not just make some assumptions about the structure of the rock?
Assume both rocks are spherically symmetric with no impurities.

 

[A.T.]

how much "damage" will the rocks receive as a function of their momentums?

Their momenta are frame dependent while the damage is frame invariant, so it doesn't makes sense to relate them.

 

[cjl]

Their momenta are frame dependent while the damage is frame invariant, so it doesn't makes sense to relate them.

The difference in their momenta should be frame invariant though.

 

[A.T.]

The difference in their momenta should be frame invariant though.

Not in general.

 

[bluecheez]

Whether or not the rocks will be damaged will depend on a lot more than just their momenta. It will depend on their structure and shape, the point and angle of impact, impurities in the structure of the rock, small existing cracks in the rocks etc.

This is not something you can derive from basic physics equations or momenta alone.

Their momenta are frame dependent while the damage is frame invariant, so it doesn't makes sense to relate them.

So what stops you from just fixing your reference frame? Sure if it makes your life easier you can change the reference frame. But I don't see how allowing someone the freedom to choose the reference frame makes the problem "not make sense."

 

[A.T.]

But I don't see how allowing someone the freedom to choose the reference frame makes the problem "not make sense."

A different reference frame will see different momenta, but still the same damage. So it doesn’t make sense to look for "damage as a function of their momenta".

 

[bluecheez]

A different reference frame will see different momenta, but still the same damage. So it doesn’t make sense to look for "damage as a function of their momenta".

So what's wrong with:
"damage as a function of their momenta at a specific, specified reference frame"?

 

[A.T.]

So what's wrong with:
"damage as a function of their momenta at a specific, specified reference frame"?

You didn't specify a reference frame.

 

[bluecheez]

You didn't specify a reference frame.

So what stops you from just fixing your reference frame? Sure if it makes your life easier you can change the reference frame.

I mean I would prefer to leave the reference frame up to the person doing the problem so that they can make the calculation easier. But if you really NEED me to pick a reference frame before even thinking about the problem, then let's say that it's in the frame where the momentum of the first mass is zero.

 

[BOYLANATOR]

I think rather than thinking about the momentum you should think about the center of mass frame and think of another physical quantity that is more associated with the "damage".

 

[bluecheez]

I think rather than thinking about the momentum you should think about the center of mass frame and think of another physical quantity that is more associated with the "damage".

Like what physical quantity?

 

[BOYLANATOR]

Kinetic Energy. In the center of mass frame the rocks will collide with maximum "damage" which will be directly proportional to kinetic energy, not momentum. The energy turns into heat, sound, can break the bonds that hold the rock together and fire the pieces in different directions.

 

[cjl]

Not in general.

Can you give me a numerical example where it wouldn't be? Off the top of my head, it seems like if you have two rocks, A and B, with some associated momenta P_A and P_B, |P_B-P_A| should be frame invariant, and P_B-P_A should also be frame invariant so long as you keep in mind any necessary coordinate rotation.

 

[Nugatory]

Can you give me a numerical example where it wouldn't be? Off the top of my head, it seems like if you have two rocks, A and B, with some associated momenta P_A and P_B, |P_B-P_A| should be frame invariant, and P_B-P_A should also be frame invariant so long as you keep in mind any necessary coordinate rotation.

Consider a 1000 kilogram elephant struck by a .1 kg bullet fired from an elephant rifle with a speed of 1 km/sec. Using coordinates in which the elephant is at rest, the momentum of the elephant is zero, the momentum of the bullet is 100 kg-m/sec, and the difference is 100 kg-m/sec.

Using coordinates in which the bullet is at rest and being swatted by an elephant moving towards it at at 1 km/sec, the momentum of the elephant is 10⁶ kg-m/sec, the momentum of the bullet is zero, and the difference is 10⁶ kg-m/sec.

This is why wise people use center-of-mass coordinates in which the total momentum is zero, unless the masses are so different (as they are when you're shooting an elephant) that we can treat the more massive object as stationary before and after the collision.

 

[cjl]

Ahh - that makes sense. For some reason, I wasn't thinking correctly about that, and you're definitely correct. I do agree that for collisions, the CoM frame is usually convenient.