Validity of Elastic Collision Equations in Two and Three Dimensions

[Karol]

Homework Statement

In a one dimensional elastic collision there are two equations: conservation of momentum and conservation of energy. by solving those 2 sets we get that the relative velocity before equals minus the relative velocity after:
v_1-v_2=-(u_1-u_2)
Is this equation valid also in the general case of two and three dimensional collisions?
i.e. is it: $\mathbf v_1-\mathbf v_2=-(\mathbf u_1-\mathbf u_2)$
I have difficulties trying to solve an example i made up and i will post it separately

 

[BvU]

In the center of mass, a collison of two objects is one-dimensional if we do not take the extent (size, shape) of the participants into consideration.

 

[Karol]

Good, but this equation derived in the laboratory coordinate system, i read it in a book.
I understand that this equation is good for the center of mass if it's a three dimensional problem, and also for the laboratory system if it's a one dimensional, right?

 

[dauto]

Yes that's correct. The logic goes like this
* The collision in the CM frame is always 1-D so the theorem always applies in the CM frame.
* The coordinate transform that relates the CM frame with the Lab frame doesn't affect relative velocities so the theorem always applies to the Lab frame as well.

 

[Karol]

* The collision in the CM frame is always 1-D

What is 1-D? is D the two or three dimensions? and if it is 1-D is negative.
I don't understand

 

[dauto]

1-D is not an equation. It means one dimensional.

 

[Karol]

Even if the collision is 3-D is still the collision in the CM 1-d? i think it's 2-D, no?
Do all 3-D collisions transform to 1-D in the center of mass? it's impossible
I mean of course they are 1-D but the plane of the movement changes, doesn't it matter? does the equation:
$v_1-v_2=-(u_1-u_2)$
Hold in CM even if the plane changes?

 

[dauto]

Even if the collision is 3-D is still the collision in the CM 1-d? i think it's 2-D, no?
Do all 3-D collisions transform to 1-D in the center of mass? it's impossible
I mean of course they are 1-D but the plane of the movement changes, doesn't it matter? does the equation:
$v_1-v_2=-(u_1-u_2)$
Hold in CM even if the plane changes?

Yes you're correct. I miss-stated the dimension. That doesn't affect the logic which is that the theorem is always valid in the CM frame so it must always be valid in any frame.

 

[Karol]

But in CM it's 1-D, is it valid in Lab 3-D?

 

[dauto]

Yes it is valid in any frame

 

[BruceW]

Homework Statement

In a one dimensional elastic collision there are two equations: conservation of momentum and conservation of energy. by solving those 2 sets we get that the relative velocity before equals minus the relative velocity after:
v_1-v_2=-(u_1-u_2)
Is this equation valid also in the general case of two and three dimensional collisions?
i.e. is it: $\mathbf v_1-\mathbf v_2=-(\mathbf u_1-\mathbf u_2)$
I have difficulties trying to solve an example i made up and i will post it separately

uh... why is there a negative sign there? I'm guessing the two objects have equal mass, which is why you have an equation with no masses in it. But I'm not sure about the negative sign. Maybe it is best to start with an equation of the form (momentum before=momentum after) and see where that takes you.

edit: also, does 1 mean before and 2 mean after? or does $u$ mean before and $v$ mean after?

 

[dauto]

You guessed wrong. The equation is always valid even if the masses are different and the sign is correct as is. v means before while u means after. 1 and 2 label the particles.

 

[dauto]

OK, what I said before really isn't entirely correct. I should've been more careful with my statements. Only the component of the vectors along the direction of the impulse satisfy the equation. The other component doesn't

 

[BruceW]

ah, OK, I see what's going on now. yeah, I agree. It is the component of the velocity in the direction of relative velocity which satisfy that equation. Also, there is the 'trivial' case where the two objects just continue along the same course. But that's not so interesting.

 

[dauto]

Note though that the component perpendicular to the impulse satisfy the same equation except for a reversed sign.

 

[BruceW]

I don't think it is possible to know the direction of the new relative velocity. Only the magnitude. Anyway, I'd say the OP'er should write down the 3d vector equations for momentum conservation and kinetic energy conservation, and maybe use the center of momentum frame of reference (for simplicity), and try to see where it takes him/her.

 

[dauto]

Yes, that's correct. It's not possible to predict the direction. That was my mistake earlier in the thread. I misread the equation as being an statement about magnitudes and said it was correct. Later on I realized it was intended as a vector equation which is not correct since the direction is unpredictable. I had to come back and retract the earlier statement.

 

[haruspex]

It is the component of the velocity in the direction of relative velocity which satisfy that equation.

No, as dauto wrote it applies in the direction of the impulse. In an oblique impact that is not the same as the line of relative velocity.

 

[BruceW]

well, that equation applies for the old components of velocity in the direction of the old relative velocity, and the new components of velocity in the direction of new relative velocity. So I think both our statements were a bit badly made. Also, it should be a plus/minus sign really. Anyway, it would be much better to write it in terms of vectors and inner products, so that's my advice to the OP'er if he returns.