This document provides a summary of key concepts in two-dimensional kinematics and projectile motion. It begins by defining displacement, velocity, and acceleration in two dimensions. It then discusses solving kinematics problems by resolving vectors into horizontal and vertical components. The document also covers projectile motion, where the horizontal velocity is constant and vertical acceleration is due to gravity. It ends by discussing relative velocity problems involving adding velocities of objects moving relative to each other or to a fixed point.

1. Newtonian Mechanics Chapter 3 Kinematics in Two Dimensions

4. Chapter 3 Motion in Two Dimensions Section 1: Displacement, Velocity, and Acceleration

6. Ahhh the memories…

7. Displacement

8. Velocity Average velocity is the displacement divided by the elapsed time.

9. Velocity The instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time.

10. Acceleration DEFINITION OF AVERAGE ACCELERATION

13. Chapter 3 Motion in Two Dimensions Section 2: Equations of Kinematics in Two Dimensions

14. Quick Review Equations of Kinematics

16. x component

17. y component

19. Example 1 A Moving Spacecraft In the x direction, the spacecraft has an initial velocity component of +22 m/s and an acceleration of +24 m/s 2 . In the y direction, the analogous quantities are +14 m/s and an acceleration of +12 m/s 2 . Find (a)  x and v x , (b)  y and v y , and (c) the final velocity of the spacecraft at time 7.0 s.

20. Example 1 A Moving Spacecraft In the x direction, the spacecraft has an initial velocity component of +22 m/s and an acceleration of +24 m/s 2 . In the y direction, the analogous quantities are +14 m/s and an acceleration of +12 m/s 2 . Find (a)  x and v x , (b)  y and v y , and (c) the final velocity of the spacecraft at time 7.0 s.  x a x v x v ox t ? +24.0 m/s 2 ? +22 m/s 7.0 s  y a y v y v oy t ? +12.0 m/s 2 ? +14 m/s 7.0 s

21. x component  x a x v x v ox t ? +24.0 m/s 2 ? +22 m/s 7.0 s

22. y component  y a y v y v oy t ? +12.0 m/s 2 ? +14 m/s 7.0 s

23. Resultant velocity

30. Chapter 3 Motion in Two Dimensions Section 3: Projectile Motion

32. Example 3 A Falling Care Package The airplane is moving horizontally with a constant velocity of +115 m/s at an altitude of 1050 m. Determine the time required for the care package to hit the ground.

33.  y a y v y v oy t -1050 m -9.80 m/s 2 0 m/s ?

34. 0  y a y v y v oy t -1050 m -9.80 m/s 2 0 m/s ?

36.  y a y v v o,y v o,x t -1050 m -9.80 m/s 2 ? 0 m/s 115 m/s 14.6 s

38. Example 6 The Height of a Kickoff A placekicker kicks a football at and angle of 40.0 degrees and the initial speed of the ball is 22 m/s. Ignoring air resistance, determine the maximum height that the ball attains.

40.  y a y v y v oy t ? -9.80 m/s 2 0 14 m/s

41. 3.3 Projectile Motion  y a y v y v oy t ? -9.80 m/s 2 0 14 m/s

42. Example 7 The Time of Flight of a Kickoff What is the time of flight between kickoff and landing?

43.  y a y v y v oy t 0 -9.80 m/s 2 14 m/s ?

44. Example 8 The Range of a Kickoff Calculate the range R of the projectile.

45. 0  x a x v x V o,x t ? 0 17 m/s 2.9 s

46. Question #9 A diver running 1.8 m/s dives out horizontally from the edge of a vertical cliff and 3.0 s later reaches the water below. How high was the cliff? h d v

47. Question #10 A diver running 1.8 m/s dives out horizontally from the edge of a vertical cliff and 3.0 s later reaches the water below. How far from its base did the diver hit the water? h d v

52. Vertical Position Equation  v o h x y Vertical Position as a function of v o ,  , and horizontal position 0

53. Maximum Height Equation for the maximum Height of a projectile. At max height, v y = 0  v o h

54. Range Equation  v o d

55. Chapter 3 Motion in Two Dimensions Section 4: Relative Velocity

57. velocity of person relative to the ground velocity of person relative to the train velocity of train relative to the ground

58. Example 11 Crossing a River The engine of a boat drives it across a river that is 1800m wide. The velocity of the boat relative to the water is 4.0m/s directed perpendicular to the current. The velocity of the water relative to the shore is 2.0m/s. (a) What is the velocity of the boat relative to the shore? (b) How long does it take for the boat to cross the river?

59.  x v BW v WS v BS t 1800m 4.0m/s 2.0m/s ? ?

60. 0  x v BW v WS t 1800m 4.0m/s 2.0m/s ?

68. END