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 mv = F) = (iii) Assume the particle starts in xo O 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 v, with = 8e-2 v = m and in this case the particle's velocity in x = 2 is 8e-2 v(2) = v² m

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 = (iii) Assume the particle starts in xo energy Eo of the particle. Using (ii), with = F) O with positive initial velocity vo > 0. Find the initial show that the particle reaches x = 2 only if vo > û, 8e-2 = " m and in this case the particle's velocity in x = 2 is 8e-2 v(2): = v 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…