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### Impulses and collisions

Posted: Fri Oct 31, 2008 9:46 pm UTC
So, if I have two completely rigid objects that collide elastically in an idealized, frictionless 2-dimensional universe, each of them goes into the collision with some linear momentum and some angular momentum, and each comes out with some linear momentum and some angular momentum.

It seems like we have six unknowns: each object's final angular momentum, each object's final momentum in the X direction, and each object's final momentum in the Y direction.

To find these 6 unknowns, I know of 5 equations of use: conservation of energy, conservation of angular momentum about each of the objects' center of gravity, conservation of linear momentum in the X direction, and conservation of linear momentum in the Y direction.

So, that's 5 equations, 6 unknowns. There must be a sixth equation tied up in how they collide, since the case where they hit is distinct from the case where they miss each other entirely, but I don't know how to extricate it. Is it going to be something beautiful, like "incoming angle equals outgoing angle," with the points of contact bouncing off of the planes tangent to the other object's surface?

### Re: Impulses and collisions

Posted: Fri Oct 31, 2008 10:48 pm UTC
Something like incoming angle equals outgoing angle is a consequence of the conservation laws, so I doubt that's it.

I think the trick may be to apply conservation of energy to both angular and linear kinetic energy if you're assuming there's no interchange between linear and angular motion.

### Re: Impulses and collisions

Posted: Fri Oct 31, 2008 11:36 pm UTC
Charlie! wrote:Something like incoming angle equals outgoing angle is a consequence of the conservation laws, so I doubt that's it.

I think the trick may be to apply conservation of energy to both angular and linear kinetic energy if you're assuming there's no interchange between linear and angular motion.

What do you mean by "no interchange?" If a spinning baseball bat hits a ball in the right way, they'll both move off in opposite direction, neither of them spinning. Does that count as interchange, even though the ball now has angular momentum about the axis the bat was spinning around?

### Re: Impulses and collisions

Posted: Fri Oct 31, 2008 11:56 pm UTC
Assuming frictionless surfaces, the 6th restriction would be that the impulse transferred be at right angles to the contact between the objects. I'm not sure how one would model friction, but that can definitely adjust the line along which the impulse happens.

Charlie! wrote:Something like incoming angle equals outgoing angle is a consequence of the conservation laws, so I doubt that's it.

I think the trick may be to apply conservation of energy to both angular and linear kinetic energy if you're assuming there's no interchange between linear and angular motion.

Incoming angle equals outgoing angle is a consequence of the conservation laws if you leave out rotational energy. In, for instance, the case of an oval hitting an oval, the impulse will generally not go through the center of gravity, and therefore you will convert from straight kinetic energy to rotational energy, and the incoming angle will generally not be the outgoing angle.

### Re: Impulses and collisions

Posted: Sat Nov 01, 2008 12:31 am UTC
Why should there be a unique solution? There isn't one for point particles: you have to know the angle of scattering of one of the particles in order to determine the rest of the variables.

In addition, you shouldn't count angular momentum conservation around two different axes. Angular momentum around one axis will automatically imply conservation around the other axis. You will have to determine some of the variables from the mechanism of the collision.

### Re: Impulses and collisions

Posted: Sat Nov 01, 2008 1:52 am UTC
ThinkerEmeritus wrote:Why should there be a unique solution? There isn't one for point particles: you have to know the angle of scattering of one of the particles in order to determine the rest of the variables.

In addition, you shouldn't count angular momentum conservation around two different axes. Angular momentum around one axis will automatically imply conservation around the other axis. You will have to determine some of the variables from the mechanism of the collision.

I'm afraid I don't get what you mean.

If I have two things hitting each other, then there will be a unique solution for that pair of things hitting each other in the way they are with the velocities they have. If I hit them together again in the same orientation with the same velocities, it will always happen the same way. (I'm ignoring quantum mechanics. My things aren't that small.)

And the equation
L1i + P2i * d2i = L1f + P2f * d2f
(L is angular momentum around center of mass, P is linear momentum, d is the shortest distance from the first object's axis to the second's velocity vector, subscript numbers designate which body's attribute it is)

is different from the equation
L2i + P1i * d1i = L2f + P1f * d1f.

Isn't it?

### Re: Impulses and collisions

Posted: Sat Nov 01, 2008 4:15 am UTC
Well, that depends on their shapes. If they are both spherical, there is no way they can affect one another's angular momentum in a frictionless universe. If they are any other shape I think you would need more than one additional constraint.

### Re: Impulses and collisions

Posted: Sat Nov 01, 2008 11:50 am UTC
Kerberos wrote:If I have two things hitting each other, then there will be a unique solution for that pair of things hitting each other in the way they are with the velocities they have. If I hit them together again in the same orientation with the same velocities, it will always happen the same way. (I'm ignoring quantum mechanics. My things aren't that small.)

And the equation
L1i + P2i * d2i = L1f + P2f * d2f
(L is angular momentum around center of mass, P is linear momentum, d is the shortest distance from the first object's axis to the second's velocity vector, subscript numbers designate which body's attribute it is)

is different from the equation
L2i + P1i * d1i = L2f + P1f * d1f.

Isn't it?

Yes, if two things hit each other in the same way, there will be a unique solution. That solution may not be determined by conservation laws, however. As Ziggy pointed out, there may be more than one way for the collision to occur, giving different results, in which case the general conservation laws had better not determine the results.

Thanks for quoting your two equations. They are indeed different, but neither one is exactly angular momentum conservation. Angular momentum is conserved in general only if the equation contains all of the angular momenta in the problem. The individual angular momenta of the two colliding objects are not necessarily unchanged by the collision; the objects can exchange angular momentum with each other. Your first equation leaves out the angular momentum of the second object, and the second leaves out the angular momentum of the first object. If you include both angular momenta in both equations, they will be equivilant.

This is a nice puzzle. I've had fun with it (so far). Thanks for bringing it up. Now if I were only still teaching mechanics....

### Re: Impulses and collisions

Posted: Sat Nov 01, 2008 1:59 pm UTC
ThinkerEmeritus wrote:Why should there be a unique solution? There isn't one for point particles: you have to know the angle of scattering of one of the particles in order to determine the rest of the variables.

In addition, you shouldn't count angular momentum conservation around two different axes. Angular momentum around one axis will automatically imply conservation around the other axis. You will have to determine some of the variables from the mechanism of the collision.

In 3 dimensions there are 3 axes along which angular momentum has to be conserved, not 2. Did you mean angular momentum conservation around two different points?

Kerberos wrote:If I have two things hitting each other, then there will be a unique solution for that pair of things hitting each other in the way they are with the velocities they have. If I hit them together again in the same orientation with the same velocities, it will always happen the same way. (I'm ignoring quantum mechanics. My things aren't that small.)

In reality, sure. But you're not dealing with reality, you're dealing with an idealization of reality. And it may be that there are an infinite set of solutions that your idealization can't tell apart that in the real world are told apart by factors that are not included in your idealization.

See viewtopic.php?f=17&t=21898&p=649419 for a discussion of one such example, an elastic collision between multiple billiard balls. In the case of 3 billiard balls there are an infinite number of possible solutions, but with real balls you tend to (unpredictably) see exactly the 6 that result from having 3 pairwise collisions really close. With 4 you can fairly easily set things up so that you get something which is not possible from having a set of pairwise collisions. This is all deterministic but determining the exact details of what happens requires understanding how the balls deform. However you can't model that exactly without modeling enough that the collision is no longer perfectly elastic. (The deformation of the ball results in energy being transformed into the shape of the ball oscillating. In the real world this oscillation dies down due to friction, and energy is lost.)

### Re: Impulses and collisions

Posted: Sat Nov 01, 2008 2:49 pm UTC
btilly wrote:
ThinkerEmeritus wrote:Why should there be a unique solution? There isn't one for point particles: you have to know the angle of scattering of one of the particles in order to determine the rest of the variables.

In addition, you shouldn't count angular momentum conservation around two different axes. Angular momentum around one axis will automatically imply conservation around the other axis. You will have to determine some of the variables from the mechanism of the collision.

In 3 dimensions there are 3 axes along which angular momentum has to be conserved, not 2. Did you mean angular momentum conservation around two different points?

The original question was specifically about a 2-dimensional problem, and I was thinking in those terms. In 3 dimensions, I agree that "point" would be a much better word than "axis."

### Re: Impulses and collisions

Posted: Sat Nov 01, 2008 9:56 pm UTC
ThinkerEmeritus wrote:Thanks for quoting your two equations. They are indeed different, but neither one is exactly angular momentum conservation. Angular momentum is conserved in general only if the equation contains all of the angular momenta in the problem. The individual angular momenta of the two colliding objects are not necessarily unchanged by the collision; the objects can exchange angular momentum with each other. Your first equation leaves out the angular momentum of the second object, and the second leaves out the angular momentum of the first object. If you include both angular momenta in both equations, they will be equivilant.

Oh, my. I, not having studied angular motion deeply, had assumed that a body rotating about its center of mass would have no angular momentum around other axes, but now that you make me think about it, that's a very silly assumption. You're completely right.

btilly: I see how having 3 or more bodies all colliding simultaneously will result in an infinite number of possibilities, but I only have two in this problem, and this doesn't require impossibly precise aim or timing. I just can't make myself believe that even knowing all of the initial conditions, it's impossible to find a uniquely correct outcome. That's how I interpreted what you were saying, anyway. If I took your post the wrong way, please tell.

### Re: Impulses and collisions

Posted: Sun Nov 02, 2008 2:10 am UTC
Kerberos wrote:btilly: I see how having 3 or more bodies all colliding simultaneously will result in an infinite number of possibilities, but I only have two in this problem, and this doesn't require impossibly precise aim or timing. I just can't make myself believe that even knowing all of the initial conditions, it's impossible to find a uniquely correct outcome. That's how I interpreted what you were saying, anyway. If I took your post the wrong way, please tell.

You took it correctly, but you're missing the role of friction. Suppose 2 balls hit, and one is spinning. If there is no friction, the spinning ball will stay spinning. If there is friction, the spinning ball will transfer some of its spin into momentum for both balls, and spin for the other ball. Therefore in your 2 ball case there must be one degree of freedom to account for how much friction there is.

Having thought about it, I think that the coefficient of friction will show up as the maximum angle from the normal that the impulse can be delivered along.

### Re: Impulses and collisions

Posted: Sun Nov 02, 2008 5:12 pm UTC
Oh, I see. Yeah. Friction.

What do you mean "the maximum angle from the normal..."? If they hit each other at all, their motions will have to change in some way, no matter how glancing the collision.

### Re: Impulses and collisions

Posted: Sun Nov 02, 2008 8:33 pm UTC
Kerberos wrote:Oh, I see. Yeah. Friction.

What do you mean "the maximum angle from the normal..."? If they hit each other at all, their motions will have to change in some way, no matter how glancing the collision.

In a frictionless impact, the impulse will always be at right angles to the surface of the impact. In the case of balls this means that angular momentum is not transferred at all. But if you have friction then the impulse will be at an angle - part of it will be the impact, and part will be the pull from friction. The angle means that the spin on the balls can change because of the impact.

### Re: Impulses and collisions

Posted: Sun Nov 02, 2008 8:59 pm UTC
Oh, gotcha.

But even frictionless, angular momentum can be transferred. If I have a rod spinning about its center of mass, then if it hits a ball moving straight towards the axis, it gives the ball angular momentum by hitting it on a course that is no longer straight through the axis. Right?

### Re: Impulses and collisions

Posted: Sun Nov 02, 2008 10:29 pm UTC
Shouldn't any frictionless collisions with a ball be perfectly frontal, and thus, transfer no angular momentum (by the way, I don't understand much of this, so I may be totally off).

### Re: Impulses and collisions

Posted: Mon Nov 03, 2008 12:21 am UTC
You're right that the ball won't have angular momentum about the axis that contains its center at any instant (if I'm wrong, someone please correct me because my understanding is very flawed), but after being hit by the rod, it will be moving on a course that does not pass through the axis of the rod's rotation. That gives it angular momentum about that axis.

### Re: Impulses and collisions

Posted: Mon Nov 03, 2008 1:49 am UTC
Kerberos wrote:Oh, gotcha.

But even frictionless, angular momentum can be transferred. If I have a rod spinning about its center of mass, then if it hits a ball moving straight towards the axis, it gives the ball angular momentum by hitting it on a course that is no longer straight through the axis. Right?

Depending on where you measure the angular momentum of the ball from, it will have angular momentum by virtue of its motion. But without friction the ball wouldn't spin.

### Re: Impulses and collisions

Posted: Mon Nov 03, 2008 2:00 am UTC
Yeah. So... what you're saying is that the angular momentum of one thing can't be directly turned into the other thing's angular momentum?

If so, how can one turn that into a more specific thing that can be used to predict what will happen?

### Re: Impulses and collisions

Posted: Mon Nov 03, 2008 3:51 am UTC
Kerberos wrote:Yeah. So... what you're saying is that the angular momentum of one thing can't be directly turned into the other thing's angular momentum?

If so, how can one turn that into a more specific thing that can be used to predict what will happen?

In the case of frictionless balls you can ignore the possibility of the balls spinning. That reduces the number of equations and variables and makes everything uniquely solvable.

In the case of other shapes without friction you can look at the spot where they contact, and model the contact as an impulse at right angles to the surface of contact, with the impact felt by the one equal to the opposite impact felt by the other. The impact will now, in general, affect both the spin and the momentum of each object. How much it affects each can be directly calculated. And you are now down to one variable that you need to solve for - the size of the impulse. However conservation of energy will give you a quadratic equation with 2 solutions. One corresponds to no impulse, and the other is the answer you want for what happens after the impact.

When you add friction you have the possibility that the impulse passes through the point of contact at a different angle from the surface of the contact. Which angle is, however, not so simple to figure out.

### Re: Impulses and collisions

Posted: Mon Nov 03, 2008 4:11 am UTC
Oh, I see what you mean now. Thanks for your help. I think I can probably finish this now.

### Re: Impulses and collisions

Posted: Wed Nov 26, 2008 7:50 pm UTC
(Should I have made a new thread after letting this one stay dead for so long? If the stink of its rotten flesh bothers people on the first page, I'd be glad to.)

I have a theory about the directions of impulses when two things collide that I'd like to check against what other people think.

It seems like if I have an object that isn't rotating, just moving, and it bangs into something else that is holding still, then what it really wants to do is deliver an impulse that is parallel to its direction of motion. Of course, if the collision is very grazing, it won't be able to do so because the frictional force can only be µ times the normal force (correct?). But with a sufficiently high coefficient of friction, it seems like this is what should happen.

And because the frictional force can only be µ times the radial force, the maximum angle the impulse can stray from the normal force should be tan-1 µ, yeah?

So if those two things are both true, then the impulse will either be delivered along a line parallel to the moving object's direction of motion or a line that is (tan-1 µ) towards the direction of motion from the normal force. Is that right?

### Re: Impulses and collisions

Posted: Tue Dec 09, 2008 4:52 am UTC
Any feedback at all would be pretty useful, really.

Also in the "does this seem reasonable" vein, I have this equation for the momentum transfer in the case of two identical disks moving in opposite directions at equal speed:
A = k * (m*R*(sin θ)*(ω12 + x•P) / ((sin θ) - k)

A: magnitude of momentum transfer
k: (moment of inertia) / (mass * radius^2) = 1/2
m: mass of 1 disk