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

Let x=x(t) be a twice-differentiable function and consider the second order differential equation x+ax+bx=0(11) Show that the change of variables y = x' and z = x allows Equation (11) to be written as a system of two linear differential equations in y and z. Show that the characteristic equation of the system in part (a) is 2+a+b=0.

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. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ŷ, with 8e-2 v= and in this case the particle's velocity in x = 2 is m (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. 7 8e-2 v(2) V (iv) Assume the particle starts in xo = 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 →→∞? m (v) Assume m = 1, show that the…

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…