Solute the question 4and 5，please H2. A particle of mass m moves in a straight line under the action of a conservative force F(x)with potential energyU(x)=x²e-x.(i) Calculate F(x) and find the two equilibrium points of the system (i.e., points xe such thatF(x) = 0). Compute if the equilibria are stable (i.e., local minima of the potential energy:U" (xe) > 0) or unstable (i.e., local maxima of the potential energy: U"(xe) <0). Sketch thepotential energy as a function of x, indicating the equilibria on your plot.(ii) Calculate the total mechanical energy E of the system (i.e., kinetic plus potential), in termsof 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 =0 with positive initial velocity vo > 0. Find the initialenergy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ŷ, with8e-2v=and in this case the particle's velocity in x = 2 ism(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(f) x = 0, vo= -10.78e-2v(2) V(iv) Assume the particle starts in xo = 0 with positive initial velocity vo > û. Use (ii) to find theexpression for v(x) and find the terminal velocity of the particle as x→∞. If the particle startswith negative initial velocity vo < 0, can it escape to x →→∞?m(v) Assume m = 1, show that the equation of motion is = x(x - 2)e¯x.Using a computing software (e.g. Python), solve this equation and plot the solutions (x as afunction of time t) for t = [0, 10] and for the six initial conditionsDiscuss the behaviour of the solutions in light of the previous points.(should you need help with the programming exercise, remember that weekly workshops areavailable for MATH1005 to help you set up your notebook file and get the code running!)

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H2. 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 (i.e., points xe such that F(x) = 0). Compute if the equilibria are stable (i.e., local minima of the potential energy: U" (xe) > 0) or unstable (i.e., local maxima of the potential energy: U"(xe) < 0). 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 (i.e., kinetic plus potential), 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) GHD 0 atarta Find th tial