Halliday, Resnick & Walker — Potential Energy and Conservation of Energy: Problem 41

41 A single conservative force \( F(x) \) acts on a \( 1.0 \text{ kg} \) particle that moves along an \( x \) axis. The potential energy \( U(x) \) associated with \( F(x) \) is given by \[ U(x) = -4x e^{-x/4} \text{ J}, \] where \( x \) is in meters. At \( x = 5.0 \text{ m} \) the particle has a kinetic energy of \( 2.0 \text{ J} \). (a) What is the mechanical energy of the system? (b) Make a plot of \( U(x) \) as a function of \( x \) for \( 0 \le x \le 10 \text{ m} \), and on the same graph draw the line that represents the mechanical energy of the system. Use part (b) to determine (c) the least value of \( x \) the particle can reach and (d) the greatest value of \( x \) the particle can reach. Use part (b) to determine (e) the maximum kinetic energy of the particle and (f) the value of \( x \) at which it occurs. (g) Determine an expression in newtons and meters for \( F(x) \) as a function of \( x \). (h) For what (finite) value of \( x \) does \( F(x) = 0 \)?