From the section COMPLETING THE CIRCLE

• 4.1:
  • a.)
    We start by doing just as instructed using Maple

    
    

    where the white curve represents the potential energy (it should be minimum at t = 0 and the ending time of about t = 10 seconds), the red curve is the kinetic energy (maximum at the two bounding times where the velocity is greatest), and the green line represents the total energy (constant since it is conserved).

  • b.)
    The kinetic energy must be zero at the greatest height of the ball since that is the point where motion of the ball momentarily ceases as it reverses direction and begins to fall. The time for which the kinetic and potential energies are equal are found as follows

    

    The times are at 1.44 seconds and 8.56 seconds.

• 4.2:
  • a.)
    We choose to have k = 5. The plot for potential energy is below

    
    

  • b.)
    Since we know that energy is conserved, we can easily determine that, at any x point, KE = TE - PE, where KE is the kinetic energy, PE is the potential energy, and TE is the total energy of the system. The numerical value for the total energy is just the potential energy at the point where the kinetic energy is zero (namely the mass is at one of the amplitude positions). Hence, for k = 5,
    

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