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

 

[PJC01]

for your question. The problem with this calculation is that it assumes a perfectly inelastic collision, where the two masses stick together after the collision. In reality, there will always be some loss of kinetic energy due to factors such as friction and deformation of the objects involved. This is why the coefficient of restitution, e, is used to account for this loss of kinetic energy.

In your calculation, you have assumed that the coefficient of restitution is 1, which would mean a perfectly elastic collision. However, in an inelastic collision, the coefficient of restitution is always less than 1. This means that some of the kinetic energy is lost in the form of heat, sound, and deformation.

In order to accurately calculate the kinetic energy loss in an inelastic collision, you would need to know the specific properties of the objects involved, such as their elasticity, surface roughness, and any other factors that could affect the collision. Without this information, it is not possible to accurately calculate the amount of kinetic energy lost.

In summary, the problem with your calculation is that it does not take into account the loss of kinetic energy in an inelastic collision. To accurately calculate this, you would need to know the specific properties of the objects involved and use a more complex equation that takes into account the coefficient of restitution and other factors.