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:
$v_1=-\frac{m_2}{m_2-m_1}v_r\\<br /> v_2=\frac{m_1}{m_2-m_1}v_r\\<br /> u_1=\frac{m_2}{m_2-m_1}u_r\\<br /> u_2=-\frac{m_1}{m_2-m_1}u_r$
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.