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I am having trouble answering the following question, please could you help! Thank you in advance for any assistance you can give.

Consider classical Rutherford scattering of a particle with mass $m$ and initial speed $v_0$ from a potential \begin{equation} V(r) = \frac{\alpha}{r} \end{equation} where $\alpha$ is some constant and $r$ is the location of the particle from the origin. Starting from Newton's second law, show that \begin{equation} |\Delta {\bf{p}}| = \frac{2 \alpha}{v_0 b} \cos \Big(\frac{\Theta}{2} \Big). \end{equation} Note that $b$ is the impact parameter and $\Theta$ is the scattering angle.

Please note that I have already shown that "from geometry" the change in momentum is $|\Delta {\bf{p}}| = 2 p \sin(\Theta / 2)$, and that $b v_0 = r^2 \frac{d \theta}{dt}$ where $t$ is time and $\theta$ is the angle $\angle({\bf{r}},{\bf{r^*}})$ where ${\bf{r^*}}$ is the point of closest approach. I am unsure however if the the above two equations will be of assistance.

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