2. A particle of mass m moves in a straight line under the action of a conservative force F(z) with potential energy (i) Calculate F'(z) 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 z, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system, in terms of v and z. 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 = 0 with positive initial velocity vg > 0. Find the initial energy FEy of the particle. Using (ii), show that the particle reaches z = 2 only if vg > v, with - 8e2 vV = . m and in this case the particle's velocity in z = 2 1s Ser? Gmwv o= Uolimr iw H (iv) Assume the particle starts in zo = 0 with positive initial velocity vg > . Use (ii) to find the expression for v(z) and find the terminal velocity of the particle as co. If the particle starts with negative initial velocity vg