Question

Need help with part b!

The potential energy of a particle moving in the (?, ?) plane is described as:
U (r) = U0 (1/2(r/b)^4 - (r/b))^2 = mgh0(1/2(r/b)^4 - (r/b)^2)
where ? = √(?^2+?^2) is the distance of the particle from the origin.

a) Create program that calculates the position and velocity of a particle as a function of time. You can use natural units by setting all constants (?0, ?, ?, ℎ0, ?) equal to 1. The program should also calculate potential and kinetic energy as a function of time. The figures show the result for a calculation with initial conditions ?⃗0 = 1.5??̂ and ?⃗0 = 0.1√?ℎ0 ?̂ over a time period of 15 s.

S. 15 0.8 0.6 10- 0.5 0.2 0.0 0 2 10 12 02 yen -LO -0.2 -0.4 -15 -1.0 0.5 1.0 1.5 2.0 0.0 wb 10 12

Reproduce the calculations for figure 1 and 2 in photo, and use the code again to calculate the path of the particle when it starts at the same initial position ?⃗0 = 1.5??̂, but at initial velocity ?⃗0 = 1.5√?ℎ0 ?̂ and ?⃗0 = 3√?ℎ0 ?̂. Describe the movement and explain what is happening.

b) Calculate analytically what initial velocity you need to use for the particle to move on a circular path when starting from the initial position ?⃗0 = √2??̂. Use your program to check that the results are correct.