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A point-like particle $A$, coming from minus spatial infinity, heads at another one, $B$, with an impact parameter of $b$. Initial momenta are $p_A$ and $p_B=0$.

They repel each other via a Yukawa potential

$$\ \ V(r_A,r_B)= +g^2\frac{e^{-\frac{|r_A-r_B|}{\lambda}}}{|r_A-r_B|} \ge 0.$$

How does the angle of the incoming particle $A$ change, relative to how it was before?

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There is a very general theory for two body scattering in, for example, Goldstein or Landau & Lifshitz. –  Michael Brown Mar 21 '13 at 22:31
@MichaelBrown: I have no idea regarding the integral in the impact parameter link and I don't know how to directly put up the equation of motion since the particle $B$ will be accelerated too giving a super nonlinear potential $\frac{\exp{(-|\Delta r(t)|/\lambda)}}{|\Delta r(t)|}$ . –  NikolajK Mar 22 '13 at 7:59
You can treat two body motion by introducing centre of mass and relative coordinates. The centre of mass motion decouples since the total momentum is conserved and you can ignore it. The relative motion is that of a particle with a reduced mass moving in a fixed potential. Try reading up on the central body problem. –  Michael Brown Mar 22 '13 at 8:08