Motion in a Plane

1. Motion in a Plane Chapter 3 Physics Chapter 3

2. Position Vector • Goes from the origin to the object – point p Physics Chapter 3

3. Average Velocity Vector • Change in displacement over change in time Physics Chapter 3

4. Instantaneous velocity vector Physics Chapter 3

5. Instantaneous velocity • In two dimensions Physics Chapter 3

6. Average acceleration vector Physics Chapter 3

7. Instantaneous acceleration vector Physics Chapter 3

8. Example • A particle has x and y coordinates (4.0 m, 2.0 m) at time t1 = 2.0 s and coordinates (7.0 m, 6.0 m) at time t2 = 2.5 s. Find the components of the average velocity and the magnitude and direction of the average velocity during this time interval. Physics Chapter 3

9. Projectile Motion • Projectile – any body given an initial velocity which then follows a path (trajectory) based on gravitational acceleration and air resistance • Thrown ball • Bullet • Dropped package Physics Chapter 3

10. Projectile motion • We will neglect air resistance • We will also neglect curvature and rotation of the earth Physics Chapter 3

11. Projectile Motion • Consists of both horizontal and vertical motion • We will break these problems into x and y components to make them easier to solve Physics Chapter 3

12. Projectile motion • Two dimensional – gravity only acts vertically • We assign y as the vertical direction • We assign x as the horizontal direction Physics Chapter 3

13. 0 0 Projectile Motion Physics Chapter 3

14. Projectile Motion • If we express initial velocity in terms of its magnitude and angle with the x-axis Physics Chapter 3

15. Projectile Motion • We can calculate the projectile’s speed and the direction of its velocity Physics Chapter 3

16. Trajectory shape • Projectiles always travel in parabolas Physics Chapter 3

17. Example • A policeman chases a thief across city rooftops. They are both running at 5 m/s when they come to a gap between buildings that is 4 m wide and has a drop of 3 m. • The thief leaps at 5 m/s at an angle of 45°. Does he clear the gap? • The policeman leaps at 5 m/s horizontally. Does he clear the gap? Physics Chapter 3

18. Example Physics Chapter 3

19. Example – the thief Physics Chapter 3

20. 0 Example – the thief Physics Chapter 3

21. Example – the thief Physics Chapter 3

22. Quadratic formula Physics Chapter 3

23. Example – the thief Physics Chapter 3

24. Example – the thief Physics Chapter 3

25. Example – the thief Physics Chapter 3

26. Example – the thief • Yes, he clears the gap Physics Chapter 3

27. Example – the policeman • Everything is the same, except that Physics Chapter 3

28. Example – the policeman Physics Chapter 3

29. Example – the policeman Physics Chapter 3

30. Example – the policeman • No, he does not clear the gap Physics Chapter 3

31. Example 2 • A rookie quarterback throws a football with an initial upward velocity component of 16.0 m/s and a horizontal component of 20.0 m/s. • How much time is required for the football to reach the highest point of the trajectory? Physics Chapter 3

32. Example 2 a) • At the highest point, vy must be zero. • Before that, it was moving up, and after that it is moving down, so it must momentarily stop at that point. Physics Chapter 3

33. Example 2 a) Physics Chapter 3

34. Example 2 b) • b)How high is this point? Physics Chapter 3

35. Example 2 b) Physics Chapter 3

36. Example 2 c) • c) How much time is required for the ball to return to its original height? Physics Chapter 3

37. Example 2 c) Physics Chapter 3

38. Example 2 c) Physics Chapter 3

39. Example 2 c) Physics Chapter 3

40. Example 2 d) • d) How does the answer to part c) compare to the answer to part a)? • It is double. When something is thrown upward, it always takes the same amount of time to go up as it does to come down. • So the total flight time (back to the original height) is always twice the time to the highest point. Physics Chapter 3

41. Example 2 e) • e) How far has the football traveled horizontally? Physics Chapter 3

42. Example 2 e) Physics Chapter 3

43. Normal acceleration • Normal means perpendicular • The component of acceleration that is perpendicular to the path • Shows change in direction of velocity Physics Chapter 3

44. Tangential acceleration • Parallel to the path • Shows change in magnitude of velocity (change in speed) Physics Chapter 3

45. Uniform Circular Motion • When an object moves around a circle with constant speed, the acceleration vector points towards the center of the circle. • The velocity is tangent to the circle. In order for the speed to remain constant, the acceleration must be towards the center of the circle. • We call this centripetal acceleration. Physics Chapter 3

46. Uniform circular motion • See Fig. 3-11 on page 63 Physics Chapter 3

47. Uniform circular motion • The period of any motion is the time it takes to make one complete cycle. It is denoted with the letter T. • In this case it’s the time to go around the circle once. Physics Chapter 3

48. Example • The radius of the earth’s orbit around the sun is 1.49 x 1011 m, and the earth travels around the sun in 365.25 days. • What is the earth’s orbital speed in m/s? • What is the acceleration of the earth toward the sun in m/s2? Physics Chapter 3