4. Modeling Particle Motion

• Math review: Taylor series expansion

• Discretization: taking time steps
• A simple example

• In-class exercise 4.1:
  • implement as a program the scheme described in the "simple example" above, with dt = 0.1
  • run it for x0 = 1, v0 = 4, a = -0.5
  • plot the trajectory (t, x) of the particle until it reaches x < 0
  • on the same graph plot the analytic trajectory x(t)
  • compare the time to reach x = 0 with the analytic (Physics I) formula -- stop the calculation when x becomes negative, then use linear interpolation to refine the numerical result before making the comparison

• C++ and Python solutions to Exercise 4.1

• Inclusion of variable acceleration
  • a numerical integration scheme
  • reminder: use of functions

• How do we know we're right? -- using energy as an indicator of accuracy

• C++ and Python templates for Exercises 4.2 and 4.3

• Exercise 4.2: simple harmonic motion

  • initial conditions: x = 0, v = V0
  • harmonic acceleration: a = -K x,  with K = 4
  • here's how to determine the exact solution to this problem

  • (a) Take V0 = 1
    • start with dt = 0.05 and run the program until t = 2π (M_PI in cmath, math.pi in Python)
    • compare the analytic and numerical solutions as functions of time
    • plot the total mechanical energy E = 0.5 v² + 0.5 K x² as a function of time
    • reduce dt by hand until "reasonable" agreement between the numerical and analytical results is obtained -- let's say, when the amplitude or the energy changes by less than 1 percent over the course of the calculation

  • C++ and Python solutions to part (a)

  • (b) Now modify the termination condition to compute numerically the period of the motion, by finding the first time t > 0 when the particle crosses from x < 0 to x >= 0. Use linear interpolation to refine your estimate.
    • repeat your calculation of the period for V0 = 0.25, 0.5, 2, 4, 8.
    • plot the period as a function of V0 for V0 = 0.25, 0.5, 1, 2, 4, 8.

  • C++ solutions to part (b):  single V0,  internal loop over all V0,  script loop over all V0
  • Python solutions to part (b):  single V0,  internal loop over all V0,  script loop over all V0

• Exercise 4.3: not-so-simple periodic motion

  • repeat the previous exercise, but using the restoring force for a real pendulum:

         a = -K sin(x)

    with  K = 4  and the same initial conditions as above.

  • (a) what is the new expression for the energy?
    • what value of dt conserves energy to 1 percent?

  • C++ solution to part (a).
    For the Python solution, just change functions acc and potential similarly in the solution to 4.2(a)

  • (b) determine the period of the motion for V0 = 0.25, 0.5, 1, 2, 4, 8.
    • can you explain the behavior for V0 = 4, 8?
    • plot the period as a function of V0.

  • C++ solution for a single value of V0
    Change functions acc and potential as in this solution to obtain the other solutions from those for Exercise 4.2(b).

• Homework #3