Vectors and Direction

1. Vectors and Direction • In drawing a vector as an arrow you must choose a scale. • If you walk five meters east, your displacement can be represented by a 5 cm arrow pointing to the east.

2. Vectors and Direction • Suppose you walk 5 meters east, turn, go 8 meters north, then turn and go 3 meters west. • Your position is now 8 meters north and 2 meters east of where you started. • The diagonal vector that connects the starting position with the final position is called the resultant.

3. Vectors and Direction • The resultant is the sum of two or more vectors added together. • You could have walked a shorter distance by going 2 m east and 8 m north, and still ended up in the same place. • The resultant shows the most direct line between the starting position and the final position.

5. Calculate a resultant vector • An ant walks 2 meters West, 3 meters North, and 6 meters East. • What is the displacement of the ant?

6. 7.1 Finding Vector ComponentsGraphically • Draw a displacement vector as an arrow of appropriate length at the specified angle. • Mark the angle and use a ruler to draw the arrow.

8. 7.1 Finding the Magnitude of a Vector • When you know the x- and y- components of a vector, and the vectors form a right triangle, you can find the magnitude using the Pythagorean theorem.

9. 7.1 Adding Vectors • Writing vectors in components make it easy to add them.

10. 7.1 Subtracting Vectors

11. 7.1 Calculate vector magnitude • A mail-delivery robot needs to get from where it is to the mail bin on the map. • Find a sequence of two displacement vectors that will allow the robot to avoid hitting the desk in the middle.

12. 7.2 Projectile Motion and the Velocity Vector • Any object that is moving through the air affected only by gravity is called a projectile. • The path a projectile follows is called its trajectory.

13. 7.2 Projectile Motion and the Velocity Vector • The trajectory of a thrown basketball follows a special type of arch-shaped curve called a parabola. • The distance a projectile travels horizontally is called its range.

15. 7.2 Projectile Motion and the Velocity Vector • The velocity vector (v) is a way to precisely describe the speed and direction of motion. • There are two ways to represent velocity. • Both tell how fast and in what direction the ball travels.

16. 7.2 Calculate magnitude Draw the velocity vector v = (5, 5) m/sec and calculate the magnitude of the velocity (the speed), using the Pythagorean theorem.

17. 7.2 Components of the Velocity Vector • Suppose a car is driving 20 meters per second. • The direction of the vector is 127 degrees. • The polar representation of the velocity is v = (20 m/sec, 127°).

18. 7.2 Calculate velocity • A soccer ball is kicked at a speed of 10 m/s and an angle of 30 degrees. • Find the horizontal and vertical components of the ball’s initial velocity.

19. 7.2 Adding Velocity Components • Sometimes the total velocity of an object is a combination of velocities. • One example is the motion of a boat on a river. • The boat moves with a certain velocity relative to the water. • The water is also moving with another velocity relative to the land.

20. 7.2 Adding Velocity Components

21. 7.2 Calculate velocity components • An airplane is moving at a velocity of 100 m/s in a direction 30 degrees NE relative to the air. • The wind is blowing 40 m/s in a direction 45 degrees SE relative to the ground. • Find the resultant velocity of the airplane relative to the ground.

22. Vy y x 7.2 Projectile Motion • When we drop a ball from a height we know that its speed increases as it falls. • The increase in speed is due to the acceleration gravity, g = 9.8 m/sec2. Vx

23. 7.2 Horizontal Speed • The ball’s horizontal velocity remains constant while it falls because gravity does not exert any horizontal force. • Since there is no force, the horizontal acceleration is zero (ax = 0). • The ball will keep moving to the right at 5 m/sec.

24. 7.2 Horizontal Speed • The horizontal distance a projectile moves can be calculated according to the formula:

25. 7.2 Vertical Speed • The vertical speed (vy) of the ball will increase by 9.8 m/sec after each second. • After one second has passed, vyof the ball will be 9.8 m/sec. • After the 2nd second has passed, vywill be 19.6 m/sec and so on.

27. 7.2 Calculate using projectile motion • A stunt driver steers a car off a cliff at a speed of 20 meters per second. • He lands in the lake below two seconds later. • Find the height of the cliff and the horizontal distance the car travels.

28. 7.2 Projectiles Launched at an Angle • A soccer ball kicked off the ground is also a projectile, but it starts with an initial velocity that has both vertical and horizontal components. *The launch angle determines how the initial velocity divides between vertical (y) and horizontal (x) directions.

29. 7.2 Steep Angle • A ball launched at a steep angle will have a large vertical velocity component and a small horizontal velocity.

30. 7.2 Shallow Angle • A ball launched at a low angle will have a large horizontal velocity component and a small vertical one.

31. 7.2 Projectiles Launched at an Angle The initial velocity components of an object launched at a velocity vo and angle θ are found by breaking the velocity into x and y components.

32. 7.2 Range of a Projectile • The range, or horizontal distance, traveled by a projectile depends on the launch speed and the launch angle.

33. 7.2 Range of a Projectile • The range of a projectile is calculated from the horizontal velocity and the time of flight.

34. 7.2 Range of a Projectile • A projectile travels farthest when launched at 45 degrees.

35. 7.2 Range of a Projectile • The vertical velocity is responsible for giving the projectile its "hang" time.

36. 7.2 "Hang Time" • You can easily calculate your own hang time. • Run toward a doorway and jump as high as you can, touching the wall or door frame. • Have someone watch to see exactly how high you reach. • Measure this distance with a meter stick. • The vertical distance formula can be rearranged to solve for time:

37. 7.2 Projectile Motion and the Velocity Vector Key Question: Can you predict the landing spot of a projectile? *Students read Section 7.2 BEFORE Investigation 7.2

38. Vx Vy y x = ? Marble’s Path t = ?

39. t= 2y g In order to solve “x” we must know “t” Y = vot – ½ g t2 vot = 0 (zero) Y = ½ g t2 2y = g t2 t2 = 2y g

40. 7.3 Forces in Two Dimensions • Force is also represented in x-y components.

41. 7.3 Force Vectors • If an object is in equilibrium, all of the forces acting on it are balanced and the net force is zero. • If the forces act in two dimensions, then all of the forces in the x-direction and y-direction balance separately.

42. 7.3 Equilibrium and Forces • It is much more difficult for a gymnast to hold his arms out at a 45-degree angle. • To see why, consider that each arm must still support 350 newtons vertically to balance the force of gravity.

43. 7.3 Forces in Two Dimensions • Use the y-component to find the total force in the gymnast’s left arm.

44. 7.3 Forces in Two Dimensions • The force in the right arm must also be 495 newtons because it also has a vertical component of 350 N.

45. 7.3 Forces in Two Dimensions • When the gymnast’s arms are at an angle, only part of the force from each arm is vertical. • The total force must be larger because the vertical component of force in each arm must still equal half his weight.

46. 7.3 Forces and Inclined Planes • An inclined plane is a straight surface, usually with a slope. • Consider a block sliding down a ramp. • There are three forces that act on the block: • gravity (weight). • friction • the reaction force acting on the block.

47. 7.3 Forces and Inclined Planes • When discussing forces, the word “normal” means “perpendicular to.” • The normal force acting on the block is the reaction force from the weight of the block pressing against the ramp.

48. 7.3 Forces and Inclined Planes • The normal force on the block is equal and opposite to the component of the block’s weight perpendicular to the ramp (Fy).

49. 7.3 Forces and Inclined Planes • The force parallel to the surface (Fx) is given by Fx = mg sinθ.