Problem 6.7

A particle with kinetic energy $T$ is deflected by a spherical potential well of radius $R$ and depth $U_{0},$ i.e. by the field in which the potential energy of the particle takes the form $$U=\left\{\begin{aligned} 0 & \text { for } r>R \\ -U_{0} & \text { for } r<R, \end{aligned}\right.$$ where $r$ is the distance from the centre of the well. Find the relationship between the aiming parameter $b$ of the particle and the angle $\theta$ through which it deflects from the initial motion direction.