unj fmipa-fisika

1. Kalkulus II Teguh Budi P, M.Si Sesion #17-18 JurusanFisika FakultasMatematikadanIlmuPengetahuanAlam

2. Outline Projectile Motion 1/10/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 2

3. Vectors and Motion in Spacepart (1) © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 3 1/10/2011

4. One early use of calculus was to study projectile motion. In this section we assume ideal projectile motion: Constant force of gravity in a downward direction Flat surface No air resistance (usually) 1/10/2011 4 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

5. We assume that the projectile is launched from the origin at time t=0 with initial velocity vo. The initial position is: 1/10/2011 5 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

6. Newton’s second law of motion: Vertical acceleration 1/10/2011 6 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

7. Newton’s second law of motion: The force of gravity is: Force is in the downward direction 1/10/2011 7 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

8. Newton’s second law of motion: The force of gravity is: 1/10/2011 8 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

9. Newton’s second law of motion: The force of gravity is: 1/10/2011 9 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

10. Initial conditions: 1/10/2011 10 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

11. Vector equation for ideal projectile motion: 1/10/2011 11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

12. Vector equation for ideal projectile motion: Parametric equations for ideal projectile motion: 1/10/2011 12 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

13. Example 1: A projectile is fired at 60o and 500 m/sec. Where will it be 10 seconds later? The projectile will be 2.5 kilometers downrange and at an altitude of 3.84 kilometers. Note: The speed of sound is 331.29 meters/sec Or 741.1 miles/hr at sea level. 1/10/2011 13 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

14. The maximum height of a projectile occurs when the vertical velocity equals zero. time at maximum height 1/10/2011 14 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

15. The maximum height of a projectile occurs when the vertical velocity equals zero. We can substitute this expression into the formula for height to get the maximum height. 1/10/2011 15 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

16. 1/10/2011 16 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

17. maximum height 1/10/2011 17 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

18. When the height is zero: time at launch: 1/10/2011 18 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

19. When the height is zero: time at launch: time at impact (flight time) 1/10/2011 19 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

20. If we take the expression for flight time and substitute it into the equation for x, we can find the range. 1/10/2011 20 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

21. If we take the expression for flight time and substitute it into the equation for x, we can find the range. Range 1/10/2011 21 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

22. The range is maximum when is maximum. Range is maximum when the launch angle is 45o. Range 1/10/2011 22 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

23. If we start with the parametric equations for projectile motion, we can eliminate t to get y as a function of x. 1/10/2011 23 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

24. If we start with the parametric equations for projectile motion, we can eliminate t to get y as a function of x. This simplifies to: which is the equation of a parabola. 1/10/2011 24 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

25. If we start somewhere besides the origin, the equations become: 1/10/2011 25 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

26. Example 4: A baseball is hit from 3 feet above the ground with an initial velocity of 152 ft/sec at an angle of 20o from the horizontal. A gust of wind adds a component of -8.8 ft/sec in the horizontal direction to the initial velocity. The parametric equations become: 1/10/2011 26 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

27. In real life, there are other forces on the object. The most obvious is air resistance. If the drag due to air resistance is proportional to the velocity: (Drag is in the opposite direction as velocity.) Equations for the motion of a projectile with linear drag force are given on page 546. You are not responsible for memorizing these formulas. 1/10/2011 27 © 2010 Universitas Negeri Jakarta | www.unj.ac.id |

28. Thank You 1/10/2011 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | 28