Newtonian Mechanics

1. Newtonian Mechanics Chapter 3 Kinematics in Two Dimensions

2. Learning ObjectivesNewtonian Mechanics-Kinematics I.A.2. Motion in two dimensions, including projectile motion • Students should be able to add, subtract, and resolve displacement and velocity vectors, so they can: • Determine components of a vector along two specified, mutually perpendicular axes. • Determine the net displacement of a particle or the location of a particle relative to another. • Determine the change in velocity of a particle or the velocity of one particle relative to another. • Students should understand the motion of projectiles in a uniform gravitational field, so they can: • Write down expressions for the horizontal and vertical components of velocity and position as functions of time, and sketch or identify graphs of these components. • Use these expressions in analyzing the motion of a projectile that is projected with an arbitrary initial velocity.

3. Table of Contents 3.1 Displacement, Velocity and Acceleration 3.2 Equations of Kinematics in Two Dimensions 3.3 Projectile Motion 3.4 Relative Velocity

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

5. Motion in Two Dimensions • How do we describe an object moving across a surface (x & y directions)? • Similar to how we described motion along a line. • Define a reference point, axes of measurement, and positive directions. • Each Dimension is independent of the other • Think Etch-a-Sketch

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

11. Question #1 A truck drives due south for 1.2 km in 1.5 minutes. Then, the truck turns and drives 1.2 km due west for 1.5 minutes. Which one of the following statements is correct? a) The average speed for the two segments is the same. The average velocity for the two segments is the same. b) The average speed for the two segments is not the same. The average velocity for the two segments is the same. c) The average speed for the two segments is the same. The average velocity for the two segments is not the same. d) The average speed for the two segments is not the same. The average velocity for the two segments is not the same.

12. Question #2 A ball is rolling down one hill and up another as shown. Points A and B are at the same height. How do the velocity and acceleration change as the ball rolls from point A to point B? a) The velocity and acceleration are the same at both points. b) The velocity and the magnitude of the acceleration are the same at both points, but the direction of the acceleration is opposite at B to the direction it had at A. c) The acceleration and the magnitude of the velocity are the same at both points, but the direction of the velocity is opposite at B to the direction it had at A. d) The horizontal component of the velocity is the same at points A and B, but the vertical component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B is the same. e) The vertical component of the velocity is the same at points A and B, but the horizontal component of the velocity has the same magnitude, but the opposite sign at B. The acceleration at points A and B has the same magnitude, but opposite direction.

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

14. Quick Review Equations of Kinematics

15. Keys to Solving 2-D Problems • Resolve all vectors into components • x-component • Y-component • Work the problem as two one-dimensional problems. • Each dimension can obey different equations of motion. • Re-combine the results for the two components at the end of the problem.

16. x component

17. y component

18. Actual motion • The x part of the motion occurs exactly as it would if the y part did not occur at all, and vice versa. • Remember the bullet drop video!

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/s2. In the y direction, the analogous quantities are +14 m/s and an acceleration of +12 m/s2. Find (a) x and vx, (b) y and vy, 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/s2. In the y direction, the analogous quantities are +14 m/s and an acceleration of +12 m/s2. Find (a) x and vx, (b) y and vy, and (c) the final velocity of the spacecraft at time 7.0 s.

21. x component

22. y component

23. Resultant velocity

24. Question #3 An eagle takes off from a tree branch on the side of a mountain and flies due west for 225 m in 19 s. Spying a mouse on the ground to the west, the eagle dives 441 m at an angle of 65 relative to the horizontal direction for 11 s to catch the mouse. Determine the eagle’s average velocity for the thirty second interval. a) 19 m/s at 44 below the horizontal direction b) 22 m/s at 65 below the horizontal direction c) 19 m/s at 65 below the horizontal direction d) 22 m/s at 44 below the horizontal direction e) 25 m/s at 27 below the horizontal direction

25. Question #4 In two-dimensional motion in the x-y plane, what is the relationship between the x part of the motion to the y part of the motion? a) The x part of the motion is independent of the y part of the motion. b) The y part of the motion goes as the square of the x part of the motion. c) The x part of the motion is linearly dependent on the y part of the motion. d) The x part of the motion goes as the square of the y part of the motion. e) If the y part of the motion is in the vertical direction, then x part of the motion is dependent on the y part.

26. Question #5 Complete the following statement: In two-dimensional motion in the x-y plane, the x part of the motion and the y part of the motion are independent a) only if there is no acceleration in either direction. b) only if there is no acceleration in one of the directions. c) only if there is an acceleration in both directions. d) whether or not there is an acceleration in any direction. e) whenever the acceleration is in the y direction only.

27. a = 5.0 m/s2 a ay  ax Question #6 A roller coaster car rolls from rest down a 20o incline with an acceleration of 5.0 m/s2 How far horizontally has the coaster travelled in 10 seconds?

28. a = 5.0 m/s2 a ay  ax Question #7 A roller coaster car rolls from rest down a 20o incline with an acceleration of 5.0 m/s2 How far vertically has the coaster travelled in 10 seconds?

29. Question #8 Jackson heads east at 25 km/h for 20 minutes before heading south at 45 km/h for 20 minutes. Hunter heads south at 45 km/h for 10 minutes before heading east at 25 km/h for 30 minutes. Which driver has the greater average velocity, if either? a) Jackson b) Hunter c) They both have the same average velocity.

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

31. Projectile Motion • Something is fired, thrown, shot, or hurled near the Earth’s surface • Horizontal velocity is constant • ax = 0, vo,x = vx • Vertical velocity is accelerated • ay = g = 9.80665 m/s2≈ 10 m/s2 (constant) • Air resistance is ignored • If launch angle is not horizontal or vertical • must resolve initial velocity into x and y components

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.

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35. Example 4 The Velocity of the Care Package A plane is moving horizontally with a constant velocity of 115 m/s at an altitude of 1050 m. Ignoring air resistance, determine the velocity of a package just before it hits the ground.

37. Conceptual Example 5I Shot a Bullet into the Air… Suppose you are driving a convertible with the top down. The car is moving to the right at constant velocity. You point a rifle straight up into the air and fire it. In the absence of air resistance, where would the bullet land – (a) behind you (b) ahead of you (c) in the barrel of the rifle

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.

41. 3.3 Projectile Motion

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

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

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46. Question #9 v h d 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?

47. v h d 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?

48. Question #11 When a football in a field goal attempt reaches its maximum height, how does its speed compare to its initial speed? a) It is zero. b) It is equal to its initial speed. c) It is greater than its initial speed. d) It is less than its initial speed.

49. Question #12 The Zambezi River flows over Victoria Falls in Africa. The falls are approximately 108 m high. If the river is flowing horizontally at 3.6 m/s just before going over the falls, what is the speed of the water when it hits the bottom? Assume the water is in freefall as it drops.

50. Question #13 An astronaut on the planet Zircon tosses a rock horizontally with a speed of 6.75 m/s. The rock falls a distance of 1.20 m and lands a horizontal distance of 8.95 m from the astronaut. What is the acceleration due to gravity on Zircon?