A proton with kinetic energy `T` and aiming parameter `b` was deflected by the Coulomb field of a stationary `Au` nucleus. Find the momentum imparted to the given nucleus as a result of scattering.

Explanation

By momentum conservation
`underset((prot on))vec(P)+underset((Au))vec(p_(i))=underset((prot on))vec(p)+underset((Au))vec(p_(f))`
Thus the momentum transferred to the gold nucleus is clearly
`Deltavec(P)=vec(P)_(f)-vec(P)_(i)=vec(P)-vec(P)`
Although the momentum transfered to the `Au` nucleus is not small, the energy associated with this recoil is quite small and its effect back on the motion of the proton can be neglected to a first approximation. Then
`Deltavec(P)=sqrt(2mT)(1-costheta)hat(i)sqrt(2mT)sin theta hat(j)`
Here `hat(i)` is the unit vector in the direction of the incident proton and `hat(j)` is normal to it on the side on which it is scattered. Thus
`|Deltavec(P)| ~~ 2sqrt(2mT) "sin"(theta)/(2)`
Or using tan `theta//2=(Ƶe^(2))/((4pi epsilon_(0))2bT)` for the proton we get
`|Deltavec(P)| ~~ 2sqrt(2mT//(1+((2bT(4 piepsilon_(0)))/(Ƶe^(2)))^(2)))`