Derivation of Mass Ratio of Elastically Colliding Particles to its Scattering Angles

Disclaimer: The following is a non-rigorous, high-school mathematics "compatible" derivation that aims to provide quick proof of equations presented in Blackett's landmark paper. The concepts I used are based on elementary conservation rules in physics taught in many pre-university physics courses. 

BACKGROUND

In 1923, Blackett published a study of "forked" alpha particle tracks observed in a cloud chamber. In the paper, he interpreted those occasional Y-shaped tracks as a consequence of elastic nuclear collision between alpha particles and nucleus of atoms present in the atmosphere. To aid his discussion, he described a convenient way to determine the masses of the target nuclei by measuring the scattering angle of the particles from the initial trajectory of the incident particle. (see the actual equation below circled in red) 

The paper provide no mathematical derivation probably because it was considered trivial to the readers then. Since another form of this equation was cited once again in his 1948 Nobel lecture, this post attempts to show how this important equation came to be using physics concepts found in high-school texts.

DERIVATION

First, this equation concerns a situation where two rigid particles are colliding: A rigid incident particle of mass m₁ (interpreted as the alpha particle) with initial velocity u is to collide elastically with a rigid stationary target particle of mass m₂ (interpreted as the nucleus of atoms present in the atmosphere) in a classical 2-dimension Cartesian reference frame. 

The collision result in change of velocity (thus momentum) for both particles where the direction of incident velocity is taken as the "reference axis" and the scattering angle of both particles is the angle it subtends to the "reference axis" respectively. I provide a diagram below illustrating the system before and after collision;

To continue the derivation, we must first assume that the initial velocity of the target particle is zero. Practically, it means the speed of the target nucleus is negligible compared to the incident particle. Case in point: as RMS speed of the target atmospheric nuclei are in the order of 500 ms^-1 while speed of alpha particles are in the order of 20,000,000 ms^-1!

The final boxed equation is simple enough to use directly from cloud chamber photograph measurements to determine the mass of nucleus collided with an alpha particle. Moreover, it is also convenient to express the kind of scattering angles as one would expect from alpha particles colliding with nucleus of mass greater or less than that of itself.