to COMP 205 Homework 10 Apr 22, 1998

Due: April 29, 1998

• Implement the Following Problem in Matlab (35 points)

  We consider the following problem: a jogger is running along his/her favorite trail on the plain in order to get his/her daily exerc ise. Suddenly, he is being attacked by a dog. The dog is running with a constant velocity w. Compute the orbit of the dog.

  The orbit of the dog has the property that the velocity vector of the dog points at every time to its goal: the jogger. We assume that the jogger is running on some trail and that his/her movement is described by the two functions and

  Let us assume that for t = 0 the dog is at the point , and that at time t his/her position will be at . The following equations hold:

  1. : The dog is running with constant velocity.
  2. The vector of velocity of the dog is parallel to the difference vector between the position of the jogger and the dog:

     

  If we substitute in the first equation the expressions for and from the second equation, we obtain

  

  Solve this equation for and get the equation for the orbit of the dog as:

  

  Use the Matlab differential solvers ode23() or ode45() to solve for these equations. This system will have a singularity when the dog reaches the jogger. In this case, we should terminate the integration when the dog reaches the jogger. Furthermore, you should try different jogger functions like

  • . Let w = 10 (the velocity of the dog).
  • If t < 6, else . In this case the jogger turns back returns home. Again use w = 10.
  • The jogger is running along an ellipse. The function is . Initially use w = 19. In this case the dog should catch the jogger at time . Then consider an old, slow dog with w = 10.

  For each case, you should submit the Matlab program and the plots showing the trajectory of the jogger and the dog.

• Problems (from Golub/Ortega's book)
  • Page 79, Exercise 3.1.4 (15 points)
  • Page 79, Exercise 3.1.7 (10 points)

  • Page 80, Exercise 3.1.10 (10 points)

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