A particle of mass m moves in a potential given by:

\(V(x)=-V_{0}a^{2}(a^{2}+x^{2})/8a^{4}+x^{4}\)

a) Sketch V(x) and F(x)

b) Discuss the motions which may occur. Locate all equilibrium points and determine the frequency of motion about any that are stable.

c) A particle starts at a great distance from the potential well with a velocity v0 towards the well. As it passes the point x=a, it suffers a collision with another particle during which it loses a fraction (%u03B1) of its kinetic energy. How large must %u03B1 be in order that the particle is trapped in the well? How large must %u03B1 be in order that the particle by trapped in one side of the well? Find the turning points of the new motion if %u03B1=1.