Vectors and 2D Motion

1. Vectors and 2D Motion

2. Focus Question What is the difference between a scalar and a vector • Nothing they are the same thing • A scalar has direction and a vector does not • A vector has direction and a scalar does not • A vector has magnitude and a scalar does not • A scalar has magnitude and a vector does not

3. Magnitude • Line Length Larger Smaller • Relative size of arrow

4. Direction • The arrow indicates in which direction the body is acting Northeast Southeast Due West

5. More Specific Direction • Determine exact angle Original Vector Triangle Representation

6. Components • The vector can now be broken down into components or separate x and y values • X values are horizontal • Y values are vertical y x

7. X-component • Deals exclusively with horizontal • Can be calculated using X = rcos(θ) R = resultant = hypotenuse

8. Y component • Deals exclusively with vertical • Can be calculated using • Y = rsin(θ) • R = resultant = hypotenuse

9. Teacher Examples • Determine the x and y components for the following • A person walks 120 m at 48 degrees North of West If the person walks for 10 seconds what is their velocity in each cardinal direction?

10. Student Example A boat moves 150 m at 27 degrees south of east A) How far south is the boat from the origin? B) If the whole trip takes 5 seconds what is the velocity in the eastward direction?

11. Combining Components Components can be combined into a single vector y x

12. Determining the Angle Use the inverse trig functions to determine the angle y x

13. Example • Determine the total displacement and direction a car travels if it goes 200 m east then 100 m North.

14. Example A boat is being rowed a directly across a river due south at 2.5 m/s. There is strong current pushing the boat due east at 1.8 m/s. If the river is 100 m wide then how far down stream will the boat end up?

15. Example – 2 Vectors • Determine the resultant vector 10 12 60 40

16. Student Example 5 12 45 d 30

17. Teacher Example A man walks 20 m at 45 deg N of E then 50 m due East followed by 35 m at 60 deg N of W. How far and what direction is he from his original starting point?

18. Why Calculate Components? Vertical and Horizontal act independently

19. Clicker Quiz: Vectors • Which includes direction? • Vectors b) Scalar c)Magnitude 2) If a ball rolls along a frictionless surface for 50 m in 8 seconds than how fast is the ball traveling? 3) Determine the time it takes for a car that goes from 0 to m/s to cover a displacement of 1450 meters

20. Student Example - B Determine the time it takes for a car that goes from 0 to m/s to cover a displacement of 1450 meters

21. Accelerated Vertical Motion Which equation do you use when? Δt = time vi= initial velocity vf= final velocity a = -9.8m/s2 Δy= displacement

22. Accelerated Vertical Motion Δy= (1/2)(vi + vf)Δt vf = vi + aΔt Δy= viΔt+ (1/2)aΔt2 vf2 = vi2 + 2aΔy

23. Dropped Vertical Questions You know two things right away: vi= 0 m/s a = -9.8m/s2

24. Teacher Example - C A rock is dropped from a cliff 100 m high. Determine its velocity as it strikes the ground

25. Student Example - C A rock is dropped from a cliff that takes 4.5 seconds to hit the ground below. What is the height of the cliff?

26. Thrown Questions You know two things right away: vi= + if thrown upwards m/s vi = - if thrown downwards m/s a = -9.8m/s2

27. Thrown Vertical Questions Things you know for entire trip a = -9.8m/s2 vi =_____ Things you know at top of path vf = ______ t = ______ a= -9.8m/s2

28. Teacher Example - B A ball is thrown straight up in the air at an initial velocity of 58 m/s Calculate the maximum height of the ball and the time the ball is in the air

29. Student Example - B A ball is thrown straight up in the air and it takes 2.5 seconds to return to the same point. How fast was the ball thrown and what is the maximum height?

30. Teacher Example - B Before the advent of starter pistols, regular pistols were used to start races. Determine whether or not this is a safe practice assuming that the muzzle velocity is 500 m/s. Why or why not?

31. Teacher Example - B Sketch the graphs for time v. displacement velocity acceleration when the object is dropped

32. Student Example - B Sketch the graphs for time v. displacement velocity acceleration when the object is thrown up

33. Teacher Example - A A ball is thrown vertically off of the top of a 100 m tall building at 15 m/s. What is the maximum height? How long will it take to hit the ground

34. Student Example - A A ball is thrown vertically off of the top of a 50 m tall building at 25 m/s. How fast will it be going?

35. Clicker Quiz • What is the instantaneous velocity of a stone as it is dropped VERTICALLY off a cliff? • 0 b) -9.8 m/s^2 c) depends on the height 2) How tall is a cliff if it takes 5 seconds to fall when dropped? 3) What is the maximum height of a rock thrown in the air with a velocity of 22 m/s?

36. Projectile Motion X has its own: velocity and displacement Y has its own: velocity, acceleration, and displacement X AND Y COMPONENTS HAVE NOTHING TO DO WITH ONE ANOTHER. THEY ARE INDEPENDENT OF EACH OTHER

37. Two Dimensional Motion Vertical and Horizontal act independently

38. Initial Horizontal Velocity Only

39. Constant Horizontal Motion Constant Horizontal Motion means that there is no acceleration. No acceleration means the only equation is v = Δx Δt

40. Teacher Example - C Determine the horizontal distance a rock will travel if it is thrown at 10 m/s and it is in the air for 15 seconds

41. Student Example - C Determine the horizontal velocity of a rock that is thrown and lands 100 m from the base of the cliff it it takes 3.5 seconds to travel

42. Teacher Example – B (Mythbusters) If a level gun fires a bullet from y meters above the ground and a second bullet is dropped from y meters above the ground then which bullet hits the ground first #1 or #2

43. Student DEMO Two balls are rolled of the table as demonstrated. Predict which will be in the air longer.

44. Time Time is the exact same in the x and y directions so it is the key to solving these problems

45. Teacher Example - B A ball is launched horizontally off a cliff that is 50 meters high at a velocity of 35 m/s. Determine how long the ball is in the air? Determine the height of the cliff

46. Teacher Example A bullet is fired horizontally off a 100 m tall cliff with a velocity of 100 m/s. How far will the bullet land from the base of the cliff

47. Teacher Example IN the previous example what is the velocity of the bullet as it strikes the ground?

48. Student Example - B • What information do I need to determine the distance?

49. Student Example - B • Two base ball players each throw a baseball from the same height. Player A throws at 40 m/s. Player B throws at 20 m/s. • Compare the time the ball is in the air • Compare the vertical acceleration of the ball in each • Compare the vertical velocity • Compare the horizontal velocity • Compare the horizontal distance the ball travels • Compare the vertical distance the ball travels

50. Teacher Example -A A bomber drops a bomb at an altitude of 5200 meters. Determine the horizontal range the bomb will fly if the plane was travelling at 100 m/s when the bomb was launched