Special Relativity -- Elastic Particle Collision Algebra

[aamirza]

Homework Statement

Consider the following head-on elastic collision. Particle 1 has rest mass 2m_o, and particle 2 has rest mass m_o. Before the collision, particle 1 movies toward particle 2, which is initially at rest, with speed u (= 0.600c ). After the collision each particle moves in the forward direction with speeds of u₁ and u₂, respectively.

a) Apply the laws of conservation of total energy (or, equivalently, of relativistic mass) and of relativistic momentum to this collision and solve the resulting equation to find u₁ and u₂ (the resulting speeds of the two particles).

Homework Equations

Well, I know the mass is conserved in elastic collisions.

$m_1\gamma_1 + m_2\gamma_2 = m_1\gamma_{1}^{'} + m_2\gamma_{2}^{'}$

where gamma is the Lorrentz factor. I also know energy is conserved, (E_i = E_f, where E = M_oc² + (M - M_o)c² where M_ois the rest mass and M is the relativistic mass), but that basically reduces to the same thing as the mass equation.

I also know the equation for the conservation of momentum,

$p_1 + p_2 = p_{1}^{'} + p_{2}^{'}$

where

$p = M_o\gamma u$

M_o is the rest mass and u is the speed.

The Attempt at a Solution

So first I plugged in the values of the masses and speeds into the mass-conservation equation and got

$2m_o\gamma_1 + m_o\gamma_2 = 2m_o\gamma_1^{'} + m_o\gamma_2^{'}$

$3.5 - 2\gamma_1^{'} = \gamma_2^{'}$

but when I got around to plugging the numbers into the momentum equation, I got

$2m_o\gamma_1u_1 = 2m_o\gamma_{1}^{'}u_{1}^{'} + m_o\gamma_{2}^{'}u_{2}^{'}$

$2\gamma_1u_1 = 2\gamma_{1}^{'}u_{1}^{'} + \gamma_{2}^{'}u_{2}^{'}$

The problem arises when, no matter what I substitute in for any of the Lorrentz factors (gammas), I always get two unkonws in the equation, u₁ prime and u₂ prime. I can't think of a way to isolate just for one. Any help?

 

[mfb]

The concept of relativistic mass is not used any more in physics.

You have two equations, it is fine to have two unknowns.
$\gamma_1'$ and $u_1'$ have a known relation, you can use one of them to express the other (same for 2 of course).