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bobred
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Homework Statement
A mass [tex]m[/tex] travels at 1.5 x 10^8 m s^-1 and collides with another mass [tex]m[/tex] at rest. The two masses fuse to become [tex]M[/tex] and travel away at [tex]v_c[/tex]. Find an expression for [tex]v_c[/tex] using conservation of relativistic momentum and energy.
Homework Equations
[tex]E_a+E_b=E_c[/tex] and [tex]p_a+p_b=p_c[/tex]. With b at rest [tex]p_b=0[/tex] so [tex]p_c=p_a[/tex].
[tex]E_a=\frac{mc^2}{1-\sqrt{\frac{v^2}{c^2}}}[/tex] (1)
[tex]E_b=mc^2[/tex] (2)
[tex]p_a=\frac{mv}{1-\sqrt{\frac{v^2}{c^2}}}[/tex] (3)[tex]E^{2}_{tot}=M^2c^4+p^2_cc^2[/tex] (4)
The Attempt at a Solution
Energy conservation
[tex]\frac{mc^2}{1-\sqrt{\frac{v^2}{c^2}}}+mc^2=\frac{Mc^2}{1-\sqrt{\frac{v^2_c}{c^2}}}[/tex]
Momentum conservation
[tex]\frac{mv}{1-\sqrt{\frac{v^2}{c^2}}}=\frac{Mv}{1-\sqrt{\frac{v^2_c}{c^2}}}[/tex]
Inserting the above into eqn 4
[tex]\frac{M^2c^4}{1-\frac{v^2_c}{c^2}}=M^2c^4+\frac{M^2v^2_cc^2}{1-\frac{v^2_c}{c^2}}[/tex]
Am I on the right path? I can't seem to get sensible answer for [tex]v_c[/tex]
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