A particle of rest mass m is accelerated to a kinetic energy K in a nuclear reactor. This particle is incident on a stationary target particle, also of rest mass m.(adsbygoogle = window.adsbygoogle || []).push({});

a) Show that the speed of the centre of mass (that is the speed of the frame in which the total momentum is zero) is [tex]\gamma v/(\gamma +1)[/tex] where v is the speed of the incident particle and [tex]\gamma =(1-v^2/c^2)^-1/2[/tex]. Verify that this expression reduces to the usual one in the non relativistic case when v<<c.

b) If the collision is perfectly inelastic-show by using the conservation of energy and momentum that the mass M of the resulting composite is [tex]M=m\sqrt{2(\gamma +1)}[/tex] Verify that this reduces to the usual value in the non relativistic case v<<c

I have the solution to this problem, but there are a couple of things I don't understand. How do you get the following expressions

[tex] (Ptot-Vcm*Etot/c^2)/\sqrt(1-(Vcm)^2/c^2)=0 [/tex]

and

[tex]

\gamma*m*v=MVcm/\sqrt(1-(Vcm)^2/c^2)=\gamma*M*v/((\gamma+1)*\sqrt((1-\gamma^2*v^2/c^2)/(\gamma+1)^2)

[/tex]

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# Modern physics question

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