Interaction between potential wells

  I) Consider a particle of mass m, confined to follow a one dimensional path in a potential well: (x)= m(3x2/T2 — 2 x3Rar2 )) where a is a distance, indicating a size of the system (measured in meters), and T is some time constant (measured in seconds). a. Show that the units are correct, and make a sketch of the potential well. b. If this well represented the gravitational potential energy for the particle sliding (without friction) along a track, determine the equation of the track. y = y(x). c. If the particle starts at the minimum at x = 0 with a very small kinetic energy (so that it slides back and forth in the well), what is the frequency of these oscillations? d. If the particle starts at the minimum at x = 0 , how much initial velocity does it need to escape the potential well? e. If the particle starts with twice the necessary energy. how long does it take to get over the bump? There are two answers for this, depending on the initial direction of the velocity. Provide both answers. Feel free to use Mathematica, or any other source to do the definite integrals.