In the Solar neighborhood, the Milky Way has a flat rotation curve, with V(r)= Vc where Vc is a constant, implying a mass desnity profile ρ(r) ~ r^-2
Assume there is a cutoff radius R beyond where the mass density is zero. Prove that the velocity of escape from the galaxy from any radius r<R is:
Ve^2= 2Vc^2(1+ln R/r)
Integral needs to be done in two parts
The Attempt at a Solution
The 1/2 mv^2 provides the energy needed to do the work of moving the mass m against the force of gravity from a radius r to infinity.
I believe the integral needs to be evaluated at both r and R, however, I do not know what equation to integrate because integrating 1/2 mv^2 doesn't seem like it will yield the above equation.