Answer: Find Relativistic Collision Momentum & Energy

[bobred]

Homework Statement

A mass $$m$$ travels at 1.5 x 10^8 m s^-1 and collides with another mass $$m$$ at rest. The two masses fuse to become $$M$$ and travel away at $$v_c$$. Find an expression for $$v_c$$ using conservation of relativistic momentum and energy.

Homework Equations

$$E_a+E_b=E_c$$ and $$p_a+p_b=p_c$$. With b at rest $$p_b=0$$ so $$p_c=p_a$$.

$$E_a=\frac{mc^2}{1-\sqrt{\frac{v^2}{c^2}}}$$ (1)
$$E_b=mc^2$$ (2)

$$p_a=\frac{mv}{1-\sqrt{\frac{v^2}{c^2}}}$$ (3)$$E^{2}_{tot}=M^2c^4+p^2_cc^2$$ (4)

The Attempt at a Solution

Energy conservation
$$\frac{mc^2}{1-\sqrt{\frac{v^2}{c^2}}}+mc^2=\frac{Mc^2}{1-\sqrt{\frac{v^2_c}{c^2}}}$$

Momentum conservation
$$\frac{mv}{1-\sqrt{\frac{v^2}{c^2}}}=\frac{Mv}{1-\sqrt{\frac{v^2_c}{c^2}}}$$

Inserting the above into eqn 4

$$\frac{M^2c^4}{1-\frac{v^2_c}{c^2}}=M^2c^4+\frac{M^2v^2_cc^2}{1-\frac{v^2_c}{c^2}}$$

Am I on the right path? I can't seem to get sensible answer for $$v_c$$

 

[vela]

Try using the relation v/c = pc/E.

 

[bobred]

Got it thanks

$$p_c=\frac{Mv_c}{1-\sqrt{\frac{v^2_c}{c^2}}}$$ (1)

$$E_c=\frac{Mc^2}{1-\sqrt{\frac{v^2_c}c^2}}}$$ (2)

Dividing (1) by (2)

$$\frac{p_c}{E_c}=\frac{v_c}{c^2}$$ so $$v_c=\frac{E_cc^2}{p_c}$$

James