2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = x²e-x (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system, in terms of v and x. Show that dE/dt =0, i.e., the total energy is constant during motion. (hint: use the equation of motion mi = F) (iii) Assume the particle starts in x = 0 with positive initial velocity vo >0. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo> û, with 8e-2 v = m and in this case the particle's velocity in x = 2 is 8e-2 v(2) = ਪੰਨੇ m (iv) Assume the particle starts in x0 = 0 with positive initial velocity vo > û. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x → ∞. If the particle starts with negative initial velocity vo <0, can it escape to x → ∞? (v) Assume m=1, show that the equation of motion is = x(x-2)e. Using a computing software (e.g. Python), solve this equation and plot the solutions (x as a function of time t) for t = [0, 10] and for the six initial conditions (a) x = 0, vo= 0 (b) x = 2, vo = 0 (c) x = 0, vo = 0.5 (d) x = 0, vo = -0.5 (e) x=0, vo= 2 3 (f) x = 0, vo= -10. Discuss the behaviour of the solutions in light of the previous points.