- #1

Trying2Learn

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May I ask about spatial collisions of bodies?

In undergraduate dynamics, we study that when two particles college, we have two final unknowns: the final velocity of each particle.

We first use the conservation of linear momentum.

However, we supplement the analysis with the coefficient of restitution (and how it relates relative velocities) and can now solve for the final velocities of each body (we have our second equation)

Good. All clear.

Now turn to an analysis for bodies, not particles.

There are 12 final unknowns: the three final linear velocities and angular velocities of each body.

As I look at the undergraduate textbooks (Hibbeler , Beer and Johnson) they all assert the problem is complicated because the coefficient of restitution depends on the nature of the specific problem. I get that.

But may I proceed with this question by assuming we have such a coefficient?

Those same books then do a spatial collision problem where one of the bodies pivots around a hinge. I suppose that simplifies the problem (the pivot, essentially reduces the number of unknowns).

But are the books "chickening" out?

Suppose I do wish to do a spatial collision of two free bodies.

What are the equations?

(Let me assume the 1-axis is directed along the path of collision. And assume a sufficiently simple geometry)

I get we have to use coefficient of restitution in that direction.

But what other "statements" can be made and converted into "equations?"

Yes, we have conservation of linear and angular momentum – I get that.

(BTW: I am not trying to actually solve this. It is not a HW. I am just trying to understand this.)

For particles, we assumed that at the peak of the impact, the two particles move with the same velocity –this helps us once we know the coefficient of restitution.

But what would one do for bodies?

For example: could we say (as we do in particle collisions), that at the extreme peak of impact, the angular momentum of each body is the same?

Surely, we cannot "assert" the angular velocities are the same. Right?

Any advice? I am sure the solution is beyond me. Maybe I will try. I just want to see the assumptions that lead to the additional equations.