IA3 - 1 Jacky Chong IB Student D1 HL Mathematics Internal...

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Unformatted text preview: 1 Jacky Chong IB Student D1 HL Mathematics Internal Assessment Type II - Colored March 12, 2010 Internal Assessment Running with Angie and Buddy In this paper, I modeled the route traced by a pet dog named Buddy as he followed his owner Angie. With the use of Trigonometry, I successfully generated correct recursive formulae: 2 1 2 1 1 1 ] ) 1 [(---- +-- - = n n n n n X Y ut n X vt X X and 2 1 2 1 1 1 ] ) 1 [( ] ) 1 [(---- +---- + = n n n n n X Y ut n Y ut n vt Y Y . I proved them logically. Finally, I applied my model on a real-life situation. Contents: Part I Introduction I.a What Is The Situation?..............................2 I.b Variables and Constraints..........................3 Part II The Recursive Formulae II.a Coordinates Of Several Points Of Buddy..............3 II.b Patterns And Proof................................5 Part III Application Of My Discrete Mathematical Model III.a Simulation Of A Variety Of u, v, t, d Values.........8 III.b Buddy Catches Up With Angie!.....................14 III.c Assumptions And Limitations Of Model.............15 Part IV Real-life Situation IV.a Conveyor System Does It Work?....................16 IV.b The New Model....................................17 Part V Conclusion......................................20 Bibliography ...........................................22 2 Part I: Introduction I.a What Is The Situation? In this paper, I will model the route traced by Buddy, a pet dog, as he is chasing his owner, Angie. The Running Route Of Buddy And Angie Figure 1 : A is Angie and B is Buddy. Angie runs towards the North direction along the straight seashore. At the beginning, Buddy is d meters to the East of Angie. Both runners start from rest (A and B ). Angie runs at a constant speed of u ms-1 . Buddy notices Angie and runs towards her at constant speed v ms-1 . In attempt to catch Angie, Buddy runs directly towards her. He changes direction once per t second(s) when he looks up to notice Angies new position (as shown in figure 1). 3 I.b Variables and Constraints There are several variables and constraints in the following investigation: The variables are the initial distance between Angie and Buddy, d ; Angies speed, u ; Buddys speed, v ; and the time Buddy changes direction once, t . The first constraint is that all variables must not be zero. If u and v are zero, Angie and Buddy are both not moving; if t is zero, Buddy is not changing direction, and he is probably not moving also (we will see why); if d is zero, Buddy has already caught Angie before they start running. All these situations are meaningless in our investigation. Other constraints will be explored later in III.c....
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IA3 - 1 Jacky Chong IB Student D1 HL Mathematics Internal...

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