# SR question

1. Oct 25, 2008

### hyperon

1. The problem statement, all variables and given/known data

Two particles of rest mass m1 and m2 are moving at velocities u1 and u2 respectively. They then collide to form a new particle of rest mass m moving at velocity u. Show that

$$m^2=m_1 ^2+m_2 ^2+2m_1m_2\gamma_1\gamma_2(1-\frac{u_1u_2}{c^2})$$

where $$\gamma_n=\frac1{\sqrt{1-\frac{u_n ^2}{c^2}}}$$

2. Relevant equations

3. The attempt at a solution

I tried using conservation of energy and momentum to relate m to m1 and m2, but I am unable to get rid of u in the expression.

1: $$\frac{mc^2}{\sqrt{1-\frac{u^2}{c^2}}}=\gamma_1m_1c^2+\gamma_2m_2c^2$$

2: $$\frac{mu}{\sqrt{1-\frac{u^2}{c^2}}}=\gamma_1m_1u_1+\gamma_2m_2u_2$$

From 1,

$$m^2=\gamma_1 ^2m_1^2+\gamma_2 ^2m_2^2+2m_1m_2\gamma_1\gamma_2(1-\frac{u^2}{c^2})$$

When I made u the subject of eqn 2 and substituted it into the one above, I got quite a messy expression that didn't seem anywhere near the one in the question.

I also tried working in the frame of particle 2, and obtained

3: $$\gamma_1 'm_1c^2=\gamma'mc^2$$

4: $$\gamma_1 'm_1u_1 '=\gamma'mu'$$

(where prime indicates the value in the frame of particle 2)

From 3 and 4,

$$u_1 '=u'$$

$$\frac{u_1-u_2}{\sqrt{1-\frac{u_1u_2}{c^2}}}=\frac{u-u_2}{\sqrt{1-\frac{uu_2}{c^2}}}$$

When I proceeded to solve for u, I still got quite a messy expression.

Is there a way to solve for m^2 without having to involve u? A later part of the question asks for an expression of u.