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## Homework Statement

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)

## Homework Equations

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.