Question

A single conservative force F(x) acts on a 1.7 kg particle that moves along an x axis. The potential energy U(x) associated with F(x) is given by U(x) = -3xe-x/3 where U is in Joules and x is in meters. At x = 4 m the particle has a kinetic energy of 5.2 J. (a) What is the mechanical energy of the system? (b) What is the maximum kinetic energy of the particle and (c) the value of x at which it occurs?

Answer 1

Answer:

a)  Em = 2.04 J , b) K = 2.04 J  and c) only point where U is zero is x = 0

Explanation:

In this exercise they give us the function of potential energy

       U = - 3x $e^{-x/3}$

a) they ask us for the mechanical energy for x = 4 m

Mechanical energy is the sum of kinetic energy plus potential energy

      Em = K + U

      Em = 5.2 -3 4  $e^{-4/3}$

      Em = 5.2 - 12 0.2636

      Em = 2.04 J

b) Mechanical energy is conserved in the system, there are no dissipative forces (rubbing). At the point where the kinetic energy is maximum the potential energy must be ero

       Em = K

        K = 2.04 J

c) let's look for the point where U = 0

        U = - 3x $e^{-4/3}$ = 0

The first value for this condition is x = 0 m

The other value is e (-x / 3) = 0

1 / e (-x / 3) = 0

This happens for x = infinity.

It is an indeterminate, so the only point where U is zero is x = 0