Problem with inelastic collision

[ShayanJ]

Two masses m_1 and m_2 are closing each other with speeds v_1 and v_2. The coefficient of restitution is e. Calculate the amount of kinetic energy loss after caused by the collision.
I solved it in the center of mass coordinates(v_{cm}=u_{cm}=0). The relative speed before and after the collision are v_r=-(v_1+v_2) and u_r=u_1+u_2 respectively. Using conservation of momentum, we know that m_1v_1=-m_2v_2 and m_1u_1=-m_2u_2. Solving these equations for v_1,v_2,u_1,u_2, we'll have:
<br /> v_1=-\frac{m_2}{m_2-m_1}v_r\\<br /> v_2=\frac{m_1}{m_2-m_1}v_r\\<br /> u_1=\frac{m_2}{m_2-m_1}u_r\\<br /> u_2=-\frac{m_1}{m_2-m_1}u_r<br />
Substituting the above results into m_1v_1^2+m_2v_2^2=m_1u_1^2+m_2u_2^2+2Q and using u_r=e v_r, We'll have:
Q=\frac{m_1m_2}{2} \frac{m_1+m_2}{(m_1-m_2)^2} (1-e^2) v_r^2
But as you can see, this is saying that for m_1=m_2 , Q becomes infinite which has no meaning and so something must be wrong. But I can't find what is that. What is it?
Thanks

 

[mfb]

You use two different conventions for the speeds/velocities - for the relative speed, you use them as absolute values to add them, but in the conservation of momentum, you use them as vectors (which can be negative).

It is easier to use them as velocity, then your relative speed is wrong.

 

[dauto]

Your relative speeds are wrong. They should be v_r = v₂ - v₁, and u_r = u₂ - u₁ respectively

EDIT: I see that mfb beat me to the punch

 

[ShayanJ]

You use two different conventions for the speeds/velocities - for the relative speed, you use them as absolute values to add them, but in the conservation of momentum, you use them as vectors (which can be negative).

It is easier to use them as velocity, then your relative speed is wrong.

Ohh...yeah...thanks man.
Sometimes I really think I have some problems in the basics!

Your relative speeds are wrong. They should be v_r = v₂ - v₁, and u_r = u₂ - u₁ respectively

That's when you're dealing them as vectors. When you're dealing with their components, negative signs may appear which may alter that formula.