How to Derive cos(θ) in Relativistic Elastic Collisions?

[wam_mi]

Homework Statement

Q: Consider two identical particles A and B have the same mass m in the inertial frame
R where B is at rest. The two particles collide and their trajectories after
impact are symmetrical with respect to the incident direction. Let t be the
angle between the trajectories after collision. Obtain an expression of cos t as
a function of the mass m and the kinetic energy T1 of the particle A.

Homework Equations

(i) Einstein Relation / Relativistic Energy Relation between energy and momentum
E^2 = (mc^2)^2 + (mod p)^2 *c^2

where E = energy
and p = momentum

(ii) Conservation of Energy: Energy before = Energy after

(iii) Conservation of Momentum: Momentum before = Momentum after

The Attempt at a Solution

Using Energy Conservation: T1 + mc^2 + mc^2 = Energy after

Using Momentum Conservation: ?

Since I do not know any velocities and I only know particle A has kinetic energy T1 and therefore total energy T1 + mc^2... I don't know how to complete this problem.

Also I try to do momentum conservation in the x-direction

so before the impact particle A would have p(x-component)

and after the impact particle A would have p1(x-component) = p1 *cos (t/2)

and after the impact particle B would have p2 (x-component) = p2 * cos(t/2)

Please help!

Thanks a lot guys

 

[tiny-tim]

Hi wam_mi!

(have a square-root: √ and try using the X² and X₂ tags just above the Reply box )

Using Energy Conservation: T1 + mc^2 + mc^2 = Energy after

Using Momentum Conservation: ?

Nooo … start again …

don't use T₁ + mc² … it's horrible and pointless …

use energy = m/√(1 - v²/c²) and momentum = mv/√(1 - v²/c²)