Consider the motion of a free particle, of mass m, described in polar coordinates r, θ. Let its speed be v₀. Let the particle start at t = 0 at r = rᵢₙ, θ = θᵢₙ, and the distance of the closest approach to the origin be r₀.
Show that if the particle is coming in toward the origin (i.e. r is decreasing) that:

dr/dt = -v₀ [ 1 -r₀²/r² ]^(1/2)