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Two masses [itex] m_1 [/itex] and [itex] m_2 [/itex] are closing each other with speeds [itex] v_1 [/itex] and [itex] v_2 [/itex]. 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([itex] v_{cm}=u_{cm}=0 [/itex]). The relative speed before and after the collision are [itex] v_r=-(v_1+v_2) [/itex] and [itex] u_r=u_1+u_2 [/itex] respectively. Using conservation of momentum, we know that [itex] m_1v_1=-m_2v_2 [/itex] and [itex] m_1u_1=-m_2u_2 [/itex]. Solving these equations for [itex] v_1,v_2,u_1,u_2 [/itex], we'll have:

[itex]

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

[/itex]

Substituting the above results into [itex] m_1v_1^2+m_2v_2^2=m_1u_1^2+m_2u_2^2+2Q [/itex] and using [itex] u_r=e v_r [/itex], We'll have:

[itex]Q=\frac{m_1m_2}{2} \frac{m_1+m_2}{(m_1-m_2)^2} (1-e^2) v_r^2[/itex]

But as you can see, this is saying that for [itex]m_1=m_2[/itex] , 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

I solved it in the center of mass coordinates([itex] v_{cm}=u_{cm}=0 [/itex]). The relative speed before and after the collision are [itex] v_r=-(v_1+v_2) [/itex] and [itex] u_r=u_1+u_2 [/itex] respectively. Using conservation of momentum, we know that [itex] m_1v_1=-m_2v_2 [/itex] and [itex] m_1u_1=-m_2u_2 [/itex]. Solving these equations for [itex] v_1,v_2,u_1,u_2 [/itex], we'll have:

[itex]

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

[/itex]

Substituting the above results into [itex] m_1v_1^2+m_2v_2^2=m_1u_1^2+m_2u_2^2+2Q [/itex] and using [itex] u_r=e v_r [/itex], We'll have:

[itex]Q=\frac{m_1m_2}{2} \frac{m_1+m_2}{(m_1-m_2)^2} (1-e^2) v_r^2[/itex]

But as you can see, this is saying that for [itex]m_1=m_2[/itex] , 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

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