Chapter 8, Problem 41P

A single conservative force F(x) acts on a 1.0 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 m the particle has a kinetic energy of 2.0 J. (a) What is the mechanical energy of the system? (b) Make a plot of U(x) as a function of x for 0 ≤ x ≤ 10 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?