1D Elastic Collision - Velocities in the CM/ZM Frame

[Zatman]

Homework Statement

Prove that, for any 1D elastic collision between two particles: as viewed from the centre of mass (or zero momentum) frame of reference, the velocity of each particle after the collision has the same magnitude but opposite sign to its velocity before the collision.

2. The attempt at a solution
For i=1,2, let m_i, u_i, v_i be the mass, lab velocity before collision and lab velocity after the collision respectively.

The velocity of the centre of mass is:

V_{CM}=\frac{m_1u_1+m_2u_2}{m_1+m_2}=\frac{m_1v_1+m_2v_2}{m_1+m_2} (1)

Since the total momentum in the CM frame is zero, we have

m_1(u_1-V_{CM})+m_2(u_2-V_{CM})=0 (2)

m_1(v_1-V_{CM})+m_2(v_2-V_{CM})=0 (3)

Conservation of kinetic energy:

m_1u_1^2+m_2u_2^2=m_1v_1^2+m_2v_2^2 (4)

I need to show that

(i) (u_1-V_{CM})=-(v_1-V_{CM})

(ii) (u_2-V_{CM})=-(v_2-V_{CM})

I've been playing around with equations (1)-(4) (don't see much point copying my scribbles up on here) but can't seem to come up with (i) or (ii). Can anyone point me in the right direction to solving these equations? I'd very much appreciate it :)

 

[tiny-tim]

Hi Zatman!

Why aren't you using the centre of mass frame?

(ie V_CM = 0)​

 

[Zatman]

Hmm. Not sure I follow. V_CM is the velocity of the centre of mass in the lab frame (which clearly isn't zero?). Then to transform into the centre of mass frame

v_{CM}=v_{LAB}-V_{CM}

Guess I'm missing something pretty fundamental here...

 

[tiny-tim]

Prove that, … as viewed from the centre of mass … frame of reference, the velocity of each particle after the collision has the same magnitude but opposite sign to its velocity before the collision.

the question requires you to use the centre of mass frame

 

[Zatman]

I'm really confused now! How, exactly, is what I've set up not the centre of mass frame?

I've taken the velocities in the lab frame for each particle, and subtracted the lab velocity of the CM to get their respective velocities relative to the CM frame?

 

[tiny-tim]

in the centre of mass frame, m₁u₁ + m₂u₂ = 0

 

[Zatman]

I agree... ugh, that certainly throws a spanner into the works, since that would mean my velocity of the CM relative to the lab would also be zero. I thought one could find the velocity of the CM relative to the lab by differentiating

(m_1+m_2)X_{CM}=m_1x_1+m_2x_2

giving

V_{CM}=\frac{m_1v_1+m_2v_2}{m_1+m_2} ?

But if this doesn't work, how do I even transform into the CM frame?

 

[Zatman]

Wait hold on, what?

Surely, in the CM frame, it is

m_1(u_1-V_{CM})+m_2(u_2-V_{CM})

that is zero? And likewise for the velocities after collision.

 

[tiny-tim]

Surely, in the CM frame, it is

m_1(u_1-V_{CM})+m_2(u_2-V_{CM})

that is zero? And likewise for the velocities after collision.

yes

but what is V_CM in th CM frame? ​

 

[Zatman]

In the CM frame, the CM is stationary. BUT the "V_CM" in that equation is a constant equal to the velocity of the CM relative to the lab frame.

By your logic, m₁u₁ + m₂u₂=0 means that there is zero momentum in the lab frame, which isn't (necessarily) true?

 

[tiny-tim]

i'm not using the lab frame

the question requires you to use the centre of mass frame …

Prove that … as viewed from the centre of mass … frame of reference, the velocity of each particle after the collision has the same magnitude but opposite sign to its velocity before the collision.)

 

[Zatman]

I really hope I'm not coming across as awkward, but this makes no sense to me. :(

You're saying that in the CM frame m₁u₁+m₂u₂=0?

How is that possible? u₁ and u₂ are velocities that I defined in the lab frame, hence that expression is the momentum in the lab frame -- which isn't zero. When you transform it to the CM frame (which is what I thought I'd done) you get the velocities in the CM frame - then the momentum expression is zero, but this expression isn't m₁u₁+m₂u₂, it's what I wrote above? I really can't see the flaw in this reasoning.

 

[tiny-tim]

Prove that… as viewed from the centre of mass … frame of reference, the velocity of each particle after the collision has the same magnitude but opposite sign to its velocity before the collision.

How is that possible? u₁ and u₂ are velocities that I defined in the lab frame

yes

don't

work in the centre of mass frame​

 

[Zatman]

But I need to define the velocities somewhere!

Are you saying that I can just forget the lab frame altogether and consider solely the CM frame -- i.e. do no transformation?

 

[tiny-tim]

Are you saying that I can just forget the lab frame altogether and consider solely the CM frame -- i.e. do no transformation?

yes!

the question requires you to use the centre of mass frame ​

 

[Zatman]

Well, yes, I know I'm required to! It just seemed natural to set it up in a lab frame and then transform it.

Let me see what I can come up with. (thanks, tiny-tim)

 

[Zatman]

This is what I've set up in the CM frame - initial velocities u₁ and u₂, and final velocities v₁ and v₂ (NOT the same as the quantities defined in the first post!).

m_1u_1+m_2u_2=0

\Rightarrow u_1=-\frac{m_2u_2}{m_1} (1)

m_1v_1+m_2v_2=0

\Rightarrow v_1=-\frac{m_2v_2}{m_1} (2)

Kinetic energy considerations:

m_1u_1^2+m_2u_2^2=m_1v_1^2+m_2v_2^2 (3)

Now I substitute (1) and (2) into (3) to eliminate u₁ and v₁. Going through the algebra I end up with

v_2^2=u_2^2

v_2=\pm u_2

...and v₂ cannot be positive u₂ because otherwise there would be no collision?

 

[tiny-tim]

that's right

can you see now that it's much simpler working in the centre of mass frame?

 

[Zatman]

that's right

can you see now that it's much simpler working in the centre of mass frame?

Yes... when you don't over-complicate things

Thank you for your help, tiny-tim. :)