1. Lecture 3: 2-Dimensional Motion

2. 2 Launch angle: direction of initial velocity with respect to horizontal 2

3. 3 General Launch Angle

4. 4 -g v 0 Sin(  ) v 0 Cos(  ) General Launch Angle In general, v 0x = v 0 cos θ and v 0y = v 0 sin θ This gives the equations of motion:

5. 5 Range: the horizontal distance a projectile travels As before, use and Eliminate t and solve for x when y=0 (y = 0 at landing)

6. 6  Sin  Range is maximum at 45 o

7. 7 Range Gun If the range gun launches a ball 6 meters with a launch angle of 45 degrees, at which of these angles should a ball be launched to land in a bucket at 3 meters? a) 10 degrees b) 22.5 degrees c) 60 degrees d) 75 degrees e) one would also need the launch velocity of the range gun to know

8. 8 Range Gun If the range gun launches a ball 6 meters with a launch angle of 45 degrees, at which of these angles should a ball be launched to land in a bucket at 3 meters? a) 10 degrees b) 22.5 degrees c) 60 degrees d) 75 degrees e) one would also need the launch velocity of the range gun to know The range is proportional to sin2θ, so to travel half the distance, the ball would need to be launched with sin2θ = 0.5. sin(2*75 o ) = 0.5, so θ=75 o Note: there are two angles that would work [sin(2*15 o ) = 0.5 also]. How are the two solutions different? 8

9. 9 Symmetry in projectile motion

10. 10 On a hot summer day, a young girl swings on a rope above the local swimming hole. When she lets go of the rope her initial velocity is 2.25 m/s at an angle of 35.0° above the horizontal. If she is in flight for 0.616 s, how high above the water was she when she let go of the rope?

11. 11 y = (v 0 sinθ) t - 1/2 g t 2 y = 2.25 m/s * sin(35 o ) * (0.616 s) - 1/2 (9.8 m/s 2 ) (0.616 s) 2 = - 1.07 m Time to hit the water: t=0.616 Initial velocity: 2.25 m/s at 35 o above the horizontal At time t= 0.616 s, the girl is 1.07 m below her starting position, so her initial position was 1.07m above the water.

12. 12 ? Which of the three punts has the longest hang time? Punts I d) all have the same hang time abc h

13. 13 Which of the three punts has the longest hang time? d) all have the same hang time abc h The time in the air is determined by the vertical motion! Because all of the punts reach the same height, they all stay in the air for the same time. Follow-up: Which one had the greater initial velocity? Punts I

14. 14 Dropping the Ball III A projectile is launched from the ground at an angle of 30 °. At what point in its trajectory does this projectile have the least speed? a) just after it is launched b) at the highest point in its flight c) just before it hits the ground d) halfway between the ground and the highest point e) speed is always constant

15. 15 A projectile is launched from the ground at an angle of 30 º. At what point in its trajectory does this projectile have the least speed? a) just after it is launched b) at the highest point in its flight c) just before it hits the ground d) halfway between the ground and the highest point e) speed is always constant The speed is smallest at the highest point of its flight path because the y-component of the velocity is zero. Dropping the Ball III 15

16. 16 ab c) both at the same time A battleship simultaneously fires two shells at two enemy submarines. The shells are launched with the same magnitude of initial velocity. If the shells follow the trajectories shown, which submarine gets hit first ?

17. ab c) both at the same time A battleship simultaneously fires two shells at two enemy submarines. The shells are launched with the same magnitude of initial velocity. If the shells follow the trajectories shown, which submarine gets hit first ? The flight time is fixed by the motion in the y-direction. The higher an object goes, the longer it stays in flight. The shell hitting submarine #2 goes less high, therefore it stays in flight for less time than the other shell. Thus, submarine #2 is hit first. Follow-up: Did you need to know that they had the same initial speed?

18. 18 a) throw it upwards, so that it falls farthest b) throw it straight down c) 45 degrees, so that it flies the farthest d) this depends on how tall you are e) up OR down, whatever. It’s all the same. Spike it or launch it? You are standing in a large parking lot with a large walnut in your hand. Ignoring air resistance, at what angle should you throw the walnut to maximize your chance of breaking it on the pavement? (The initial speed and altitude of release are independent of angle.)

19. a) throw it upwards, so that it falls farthest b) throw it straight down c) 45 degrees, so that it flies the farthest d) this depends on how tall you are e) up OR down, whatever. It’s all the same. Spike it or launch it? You are standing in a large parking lot with a large walnut in your hand. Ignoring air resistance, at what angle should you throw the walnut to maximize your chance of breaking it on the pavement? (The initial speed and altitude of release are independent of angle.) If you throw it up, eventually it will come back to its launch altitude, and when it does it will have the same speed as you threw it with! You know this due to symmetry: it takes the same amount of time to reach the maximum altitude as it does to fall from the maximum to the launch point, so the net acceleration on the way down must equal that on the way up. (Of course, you do want to throw it up OR down, so the acceleration is maximized on contact with the pavement. The horizontal velocity doesn’t help break the nut.)

20. Monkey and the hunter: A monkey hangs from a tree at a height h from the ground. A hunter lying on the ground a distance D from the tree wants to shoot the monkey, firing just as the monkey drops. At what angle should the hunter shoot if the muzzle speed of his bullet is v 0 ?

21. Solution: For the bullet to hit the monkey, the bullet and the monkey must at some time t c be at the same place. That is, Thus,

22. 22 Relative Motion 22

23. 23 You are riding on a Jet Ski at an angle of 35° upstream on a river flowing with a speed of 2.8 m/s. If your velocity relative to the ground is 9.5 m/s at an angle of 20.0° upstream, what is the speed of the Jet Ski relative to the water? (Note: Angles are measured relative to the x axis shown.)

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25. 25 Now suppose the Jet Ski is moving at a speed of 12 m/s relative to the water. (a) At what angle must you point the Jet Ski if your velocity relative to the ground is to be perpendicular to the shore of the river? (b) If you increase the speed of the Jet Ski relative to the water, does the angle in part (a) increase, decrease, or stay the same? Explain. (Note: Angles are measured relative to the x axis shown.) 25

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27. 27 Relativity Car A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball? a) it depends on how fast the cart is moving b) it falls behind the cart c) it falls in front of the cart d) it falls right back into the cart e) it remains at rest

28. 28 A small cart is rolling at constant velocity on a flat track. It fires a ball straight up into the air as it moves. After it is fired, what happens to the ball? a) it depends on how fast the cart is moving b) it falls behind the cart c) it falls in front of the cart d) it falls right back into the cart e) it remains at rest when viewed from train when viewed from ground In the frame of reference of the cart, the ball only has a vertical component of velocity. So it goes up and comes back down. To a ground observer, both the cart and the ball have the same horizontal velocity, so the ball still returns into the cart. Relativity Car

29. 29 Relativity Cart As you stand on the platform, a train moves past you at a speed V. A man standing on the train throws a ball straight up with an initial upward speed of v 0. Describe what each of you sees. Are the two views compatible? Train view:Platform view: Ball goes straight up, and then straight down

30. 30 Relativity Car + =

31. 2-Dimensional Motion & Newton Laws of Motion (sections 5.1-5.4 )

32. 32 Kinematics: Assumptions, Definitions and Logical Conclusions Defined displacement, velocity, acceleration (also position, distance, speed)... Defined scalers (like speed) and vectors (like velocity) Laid out assumptions about free-fall noticed that 2-dimensional motion is really just two, simultaneous, 1-dimensional motions. Used this to shoot a monkey, range out a small cannon, etc. This wasn’t physics. This was preparing the language needed to talk about physics. What have we done so far?

33. 33 Newton’s Laws How can we consistently and generally describe the way objects move and interact?

34. Isaac Newton 1643-1727 Newton in a 1702 portrait by Godfrey Kneller Godfrey Kneller Newton in a 1689 portrait by Godfrey Kneller Godfrey Kneller

35. 35 Nature and nature's laws lay hid in night; God said "Let Newton be" and all was light. Newton’s epitaph, Alexander Pope I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. from a memoir by Newton

36. 36 Force Force: push or pull Force is a vector – it has magnitude and direction

37. 37 Mass Mass is the measure of how hard it is to change an object’s velocity. Mass can also be thought of as a measure of the quantity of matter in an object. 37

38. 38 Newton’s First Law of Motion If you stop pushing an object, does it stop moving? Only if there is friction! In the absence of any net external force, an object at rest will remain at rest. In the absence of any net external force a moving object will keep moving at a constant speed in a straight line. This is also known as the Law of Inertia.

39. 39 Inertia

40. 40 Newton’s First Law a) more than its weight b) equal to its weight c) less than its weight but more than zero d) depends on the speed of the puck e) zero A hockey puck slides on ice at constant velocity. What is the net force acting on the puck?

41. The puck is moving at a constant velocity, and therefore it is not accelerating. Thus, there must be no net force acting on the puck. Newton’s First Law a) more than its weight b) equal to its weight c) less than its weight but more than zero d) depends on the speed of the puck e) zero A hockey puck slides on ice at constant velocity. What is the net force acting on the puck? Follow-up: Are there any forces acting on the puck? What are they?

42. 42 a) a net force acted on it b) no, or insufficient, net force acted on it c) it remained at rest d) it did not move, but only seemed to e) gravity briefly stopped acting on it Newton’s First Law You put your book on the bus seat next to you. When the bus stops suddenly, the book slides forward off the seat. Why?

43. a) a net force acted on it b) no, or insufficient, net force acted on it c) it remained at rest d) it did not move, but only seemed to e) gravity briefly stopped acting on it The book was initially moving forward (because it was on a moving bus). When the bus stopped, the book continued moving forward, which was its initial state of motion, and therefore it slid forward off the seat. Newton’s First Law You put your book on the bus seat next to you. When the bus stops suddenly, the book slides forward off the seat. Why?

44. 44 Calibrating force Two equal weights exert twice the force of one; this can be used for calibration of a spring:

45. 45 Experiment: Acceleration vs Force Now that we have a calibrated spring, we can do more experiments. Acceleration is proportional to force:

46. 46 Experiment: Acceleration vs Mass Acceleration is inversely proportional to mass:

47. 47 Newton’s Second Law of Motion Combining these two observations gives Or, more familiarly, Acceleration is proportional to force: Acceleration is inversely proportional to mass:

48. 48 Newton’s Second Law of Motion SI unit for force Newton is defined using this equation as: 1 N is the force required to give a mass of 1 kg an acceleration of 1 m/s 2 An object may have several forces acting on it; the acceleration is due to the net force:

49. 49 Units of force: Newtons

50. The weight of an object is the force acting on it due to gravity Weight: W = F g = ma = mg vertically downwards Since, the weight of an object in Newtons is approximately 10 x its mass in kg Force of Gravity adult human 700 N ~ 160 lbs. 70 kg There is no “conversion” from kg to pounds! (Unless you specify what planet you are assuming) WHERE? Weight is not mass!

51. 51 Newton’s First and Second Laws In order to change the velocity of an object – magnitude or direction – a net force is required. (I)(I) (II) What about the bus... From the perspective of someone who didn’t know they were on the bus?

52. 52 Inertial Reference Frames Newton’s First and Second Laws do not work in an accelerating frame of reference In order to change the velocity of an object – magnitude or direction – a net force is required. (I)(I) (II) An inertial reference frame is one in which the first and second laws are true. Accelerating reference frames are not inertial. Was the bus an inertial reference frame? Is the earth an inertial reference frame? No, but acceleration due to earth’s rotation around Its axis (0.034 m/s 2 ), and due to earth’s rotation around sun (smaller) are negligible compared to g; so approximately yes. 52

53. 53 Analyzing the forces in a system Free-body diagrams: A free-body diagram shows every force acting on an object. Sketch the forces Isolate the object of interest Choose a convenient coordinate system Resolve the forces into components Apply Newton’s second law to each coordinate direction

54. 54 Free-body Diagram Example of a free-body diagram:

55. 55 Newton’s First Law a) there is a net force but the book has too much inertia b) there are no forces acting on it at all c) it does move, but too slowly to be seen d) there is no net force on the book e) there is a net force, but the book is too heavy to move A book is lying at rest on a table. The book will remain there at rest because:

56. 56 There are forces acting on the book, but the only forces acting are in the y-direction. Gravity acts downward, but the table exerts an upward force that is equally strong, so the two forces cancel, leaving no net force. Newton’s First Law a) there is a net force but the book has too much inertia b) there are no forces acting on it at all c) it does move, but too slowly to be seen d) there is no net force on the book e) there is a net force, but the book is too heavy to move A book is lying at rest on a table. The book will remain there at rest because:

57. 57 Newton’s Third Law of Motion Forces always come in pairs, acting on different objects: If object 1 exerts a force on object 2, then object 2 exerts a force – on object 1. These forces are called action-reaction pairs.

58. 58 Some action-reaction pairs 58

59. 59 Action-reaction pair? a) Yes b) No Newton’s 3 rd : F 12 = - F 21 action-reaction pairs are equal and opposite, but they act on different bodies

60. 60 Newton’s Third Law of Motion Although the forces are the same, the accelerations will not be unless the objects have the same mass. Q: When skydiving, do you exert a force on the earth? Does the earth accelerate towards you? Is the magnitude of the acceleration of the earth the same as the magnitude of your acceleration?

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62. Newton’s Third Law of Motion Contact forces: The force exerted by one box on the other is different depending on which one you push. Assume the mass of the two objects scales with size, and the forces pictured are the same. In which case is the magnitude of the force of box 1 on box 2 larger? 62

63. 63 Truck on Frozen Lake A very large truck sits on a frozen lake. Assume there is no friction between the tires and the ice. A fly suddenly smashes against the front window. What will happen to the truck? a) it is too heavy, so it just sits there b) it moves backward at constant speed c) it accelerates backward d) it moves forward at constant speed e) it accelerates forward

64. 64 When the fly hit the truck, it exerted a force on the truck (only for a fraction of a second). So, in this time period, the truck accelerated (backward) up to some speed. After the fly was squashed, it no longer exerted a force, and the truck simply continued moving at constant speed. Truck on Frozen Lake A very large truck sits on a frozen lake. Assume there is no friction between the tires and the ice. A fly suddenly smashes against the front window. What will happen to the truck? a) it is too heavy, so it just sits there b) it moves backward at constant speed c) it continuously accelerates backward d) it moves forward at constant speed e) it continuously accelerates forward Follow-up: What if the fly takes off, with the same speed in the direction from whence it came? 64

65. A 71-kg parent and a 19-kg child meet at the center of an ice rink. They place their hands together and push. (a) Is the force experienced by the child more than, less than, or the same as the force experienced by the parent? (b) Is the acceleration of the child more than, less than, or the same as the acceleration of the parent? Explain. (c) If the acceleration of the child is 2.6 m/s 2 in magnitude, what is the magnitude of the parent’s acceleration? 65

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67. On vacation, your 1300-kg car pulls a 540-kg trailer away from a stoplight with an acceleration of 1.9 m/s 2 (a) What is the net force exerted by the car on the trailer? (b) What force does the trailer exert on the car? (c) What is the net force acting on the car? 67

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