Consider two particles interacting via a repulsive central potential U(r) = k/r with k > 0. Find the minimal distance between particles, when one of them (with mass m_1) is coming from infinity with initial velocity v_0, and approaching an initially resting particle (with mass m_2) with impact parameter ρ. The impact parameter is the perpendicular distance between the initial path of an approaching particle and the target particle (see e.g. Impact Parameter on Wikipedia). Express your result in terms of k, v0, ρ, and the reduced mass.

Question

Consider two particles interacting via a repulsive central potential U(r) = k/r with k > 0. Find the minimal distance between particles, when one of them (with mass m_1) is coming from infinity with initial velocity v_0, and approaching an initially resting particle (with mass m_2) with impact parameter ρ. The impact parameter is the perpendicular distance between the initial path of an approaching particle and the target particle (see e.g. Impact Parameter on Wikipedia). Express your result in terms of k, v0, ρ, and the reduced mass.

Hint: Use center of mass coordinates, and focus on the relative motion ~r. Use conservation of energy and angular momentum for the relative coordinate: the initial angular momentum should involve the impact parameter ρ. The minimal distance between particles can be determined directly from the effective potential U_eff(r) and the energy E.

Answer 1

~r is supposed to be ~r = ~r_1 - ~r_2.