- #1

Aleolomorfo

- 73

- 4

## Homework Statement

Finding the maximum scattering angle of a particle whose mass in ##m_1## which hits with relativistic velocity ##v## a particle at rest with mass ##m_2<m_1##.

## The Attempt at a Solution

I've written the 4-momenta (p before the collision, k after the collision and the z-axis is along the direction of the incident particle):

$$p_1=(m_1\gamma,0,0,m_1\gamma v)$$

$$p_2=(m_2,0,0,0)$$

$$k_1=(E,0,\sqrt{E^2-m^2_1}\sin{\theta},\sqrt{E^2-m^2_1}\cos{\theta})$$

For ##k_2## the components are not important, it's important that ##k^2_2=m^2_2##

Then I've used the conservation of 4-momentum ##p_1+p_2=k_1+k_2##, then ##k_2=p_1+p_2-k_1##, then ##k^2_2=p^2_1+p^2_2+k^2_1+2p_1p_2-2p_1k_1-2p_2k_1##. After calculations I've found:

$$\cos{\theta}=\frac{m^2_1+m_1m_2\gamma-2E(m_1\gamma+m_2)}{m_1\gamma v\sqrt{E^2-m^2_1}}$$.

Then I've taken the derivative ##\frac{d(\cos{\theta})}{dE}## and put it equal to 0. However, I've found an equation quite difficult to solve and I think it's wrong.

I think the way I set up the problem is not incorrect, but maybe there is a easier way or some trick to reduce calcus.