Physics 3214 Unit 1 Motion. Vectors and Frames of Reference

Transcription

1 Physics 3214 Unit 1 Motion Vectors and Frames of Reference

2 Review Significant Digits 1D Vector Addition BUT First. Diagnostic QuizTime

3 Rules for Significant DigitsRule #1 All non zero digits are ALWAYS significant How many significant digits are in the following numbers? Significant Figures 5 Significant Digits 4 Significant Figures

4 Rule #2 All zeros between significant digits are ALWAYS significant How many significant digits are in the following numbers? Significant Figures Significant Digits 4 Significant Figures

5 Rule #3 All FINAL zeros to the right of the decimal ARE significant How many significant digits are in the following numbers? Significant Figures 5 Significant Digits 7 Significant Figures

6 Rule #4 All zeros that act as place holders are NOT significant Another way to say this is: zeros are only significant if they are between significant digits OR are the very final thing at the end of a decimal

7 For example How many significant digits are in the following numbers? x Significant Digit 3 Significant Digits 6 Significant Digits 2 Significant Digits 1 Significant Digit

8 Rule #5 All counting numbers and constants have an infinite number of significant digits For example: 1 hour = 60 minutes 12 inches = 1 foot 24 hours = 1 day

9 How many significant digits are in the following numbers? x Significant Digits 6 Significant Digits 2 Significant Digits 3 Significant Digits 4 Significant Digits 1 Significant Digit 4 Significant Digits

10 When adding or subtracting Note accuracy of measurements (nearest.1?.01?.001?) Answer can be no more accurate than the LEAST accurate number used to calculate it.

11 For Example: 5.50 grams grams grams

12 OR ml ml ml --> 2.4 ml

13 When multiplying or dividing You must COUNT significant figures The answer can have only AS MANY significant figures as the LEAST of the numbers used to get it

14 Example g / 5.0 ml answer: 1.0 g/ml Did you have to count sig figs? YES!

15 Here s a tougher one C/s x 60 s/min x 60 min/hr = answer: C/hr --> C/hr OR is it C/hr Note: standard conversion factors never limit significant figures-- instruments and equipment do.

16 Some Interesting Ones! m ) A kg N s B) 2.00 kg m 35 s 10m 2

17 Diagnostic Quiz Question A B C D

18 VECTOR ADDITION

19 Recall: Vectors Vectors Quantities have both magnitude and direction It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector.

20 Vectors Every Vector has two parts. A Head (Tip) and a Tail. Tail Head(Tip)

21 Vectors One Dimensional Vectors These are vectors that indicate direction in either the x or y direction, but not both. Two Dimensional Vectors These vectors have both x and y components in its direction.

22 Vectors One dimensional Vectors can use a + or sign to show direction m/s a - 45 m b m/s 2 i + 60 N h

23 Vectors One dimensional directions can be: Up, Down, Left, Right OR Cardinal Directions Navigator/Compass North, South, East, West Cartesian Direction q

24 Picking an Angle for 2-Dimensional Vectors 1. Pick the angle from the tail of the Vector. (starting point) 2. Find a Definite Direction w.r.t. the Vector (N, S, E, W). 3. Write your angle w.r.t. the Definite Direction. Directions are to be expressed as follows: [30.0 o S of E] The expected format is below. v = 25.0 m/s [30.0 o S of E]

25 Determine the direction of the vector. Determine angle from the tail of the vector from either N or E Answer: [42 o N of E] OR [48 o E of N]

26 Determine the direction of the following vectors: A) B)

27 Practice: Sketch the following vectors. Include the magnitude and direction (if necessary) on the sketch. A)25 km [N] B) 3m/s [Up]

28 Practice: Sketch the following vectors. Include the magnitude and direction (if necessary) on the sketch. C) 12 N [25 o E of N] D) 12 N [45 o S of W]

29 Scalar and Vector Quantities Some physical quantities, such as time, distance, speed, temperature, mass, density, electric charge, etc. can be by a single number with a unit. These are scalar quantities. However, there are many other important quantities have a direction associated with them and cannot be described by a single number. These are vector quantities.

30 Scalar and Vector Quantities For example, speed and velocity are kinematic quantities that have distinctly different definitions. Speed (v), being a scalar quantity, is the rate at which an object covers distance (d) (a scalar quantity). The average speed is v Speed is ignorant of direction. total distance total time v d t

31 On the other hand, velocity quantity; it is direction-aware. v is a vector Velocity is the rate at which the position changes. The average velocity is the displacement or position change (a vector quantity) per time ratio. displacement velocity total time v d t

32 In this calculation we look from the initial starting point to the FINAL position to determine the displacement. In velocity calculations we are not concerned about the journey, but only in how to we get from where we start to the final destination.

33 Consider the motion described below: A physics teacher walks 4 meters East, 2 meters South, 4 meters West, and finally 2 meters North. Even though the physics teacher has walked a total distance of meters, her displacement is meters. During the course of her motion, she has "covered 12 meters of ground" (distance = 12 m). Yet when she is finished walking, she is not "out of place" - i.e., there is no displacement for her motion (displacement = 0 m). Displacement, being a vector quantity, must give attention to direction.

34 If this motion took 45 seconds to complete calculate: A) Average Speed B) Average Velocity

35 Lets go back to the diagnostic quiz 2. Which of the following quantities is a scalar quantity? A) displacement B) acceleration C)distance D) velocity

36 Use the diagram to determine the resulting displacement and the distance traveled by the skier during these three minutes. The skier covers a: displacement of 140 m, rightward distance of (180 m m m)= 420 m

37 1. What is the average: A) speed B) velocity 2. What was her fastest speed? 3. What was her fastest velocity?

38 4. Draw the vector diagram for this situation. Include the resultant vector for displacement. Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION.

39 Lets go back to the diagnostic quiz Question # 4 Motion in Two Directions A car travels at 90.0 km/h [W] for 20.0 min and then travels at 60.0 km/h [E] for 10.0 min. 4. Based on the information in Motion in Two Directions, the average velocity of the car is A) 300 km/h [W] C) 75.0 km/h [W] B) 80.0 km/h [W] D) 40.0 km/h [W]

40 Addition of 2-Dimensional Vectors Addition of 2-Dimensional Vectors requires other methods. 1. Graphical Method 2. Mathematical or Triangle Method 3. Component Method

41 Properties of Vectors The addition of vectors yields a RESULTANT VECTOR R A B R Order of addition does not matter!

42 Vectors that are Perpendicular to each other are INDEPENDENT of each other. Ex. Boat traveling perpendicular to the current. Ex. Projectile thrown in the air.

43 Graphical Addition of Vectors R = A + B Adding vectors Graphically has certain advantages and disadvantages.

44 Graphical Addition of Vectors R = A + B Advantages 1. You can add as many vectors as you want 2. Not a mathematical method Disadvantages 1. Need a Ruler and a Protractor 2. Not the most accurate method Not a mathematical method

45 Vector Addition: (Non-collinear vectors) Mathematical or Triangle Method 1. Use a Vector Diagram to make your triangle Rough sketch. Not drawn to scale. 2. See what kind of Triangle you have Right Triangle: Use right angle trigonometry and the Pythagorean Theorem Physics 3214 Not Right Triangle: Use component method. Post secondary

46 Mathematical or Triangle Method When 2 vectors are perpendicular, use the Pythagorean theorem. RESULTANT Finish A man walks 95 km, East then 55 km, north. Calculate his RESULTANT DISPLACEMENT. Horizontal Component 95 km [E] 55 km, [N] Vertical Component c a b c a b c Resultant c km 2 2 Start The LEGS of the triangle are called the COMPONENTS

47 BUT what about the direction? DISPLACEMENT was asked for and since it is a VECTOR we should include a DIRECTION on our final answer. N of E NOTE: When drawing a right triangle that conveys some type of motion, you MUST draw your components Tip TO Tail.

48 BUT..what about the VALUE of the angle??? Just putting North of East on the answer is NOT specific enough for the direction. We MUST find the VALUE of the angle km 55 km, N q N of E 95 km,e So the COMPLETE final answer is :

49 Vector Addition App

50 Example: 1.What is the resultant displacement of a hiker that travels 9.0 km [E] then walks 4.0 km[s]? Include a vector diagram

51 2. A hiker left her tent to go to a lake. She walked 0.8 km [S], then 1.20 km [E] and 0.3 km [N]. A) Find her resultant displacement. B) If the total trip took 42 minutes what was her average: (i) Velocity (ii) speed

52 Slightly more interestngexample A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement. = 23 m, E 6 m, S 12 m, W 20 m, N = 14 m, N R Tan q q m 1 Tan (0.6087) m, E R 14 m, N q 23 m, E The Final Answer: m, 31.3 degrees NORTH or EAST

53 What if you are missing a V.C =? component? Suppose a person walked 65 m, 25 degrees East of North. What were his horizontal and vertical components? H.C. =? m The goal: ALWAYS MAKE A RIGHT TRIANGLE! To solve for components, we often use the trig functions since and cosine. adjacent side opposite side cosineq sineq hypotenuse hypotenuse adj hyp cosq opp hyp sin q adj V. C. 65cos m, N opp H. C. 65 sin m, E

54 Example A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate the plane's horizontal and vertical velocity components. H.C. =? cosineq adjacent side hypotenuse sineq opposite side hypotenuse 32 V.C. =? adj hyp cosq opp hyp sinq 63.5 m/s adj H. C. 63.5cos m / s, E opp V. C. 63.5sin m / s, S

55 Example 5000 km, E A storm system moves 5000 km due east, then shifts course at 40 degrees North of East for 1500 km. Calculate the storm's resultant displacement km 40 H.C. V.C. cosineq adj hyp cosq adjacent side hypotenuse sineq adj H. C. 1500cos km, E opp V. C. 1500sin opposite side hypotenuse opp hyp sinq km, N 5000 km km = km q R km km R Tanq q Tan (0.364) km The Final Answer: degrees, North of East

56 Practice: TEXT Page 84 # 1, 3 Page 90 #2,4,5a)b)

57 Frame of Reference Recall? A frame of reference is a place from which motion is observed. The description of any motion depends on the frame of reference of the observer.

58 Determine the speed of the ball from following frames of reference. Pitcher Catcher

59 mythbusters-frame of Reference

60 Consider tossing a ball into the air and catching it What if you were in a car moving along a road? What would it look like to a person standing on the side of the road?

61 Have you ever walked up the down escalator? Have you ever experienced a headwind and a tailwind? Cross wind? Where would this be important?

62

63

64

65

66

67

68

69

70 Physics Type Problems A plane (or related object) experiencing wind perpendicular to its velocity (or at a crosswind at 90 O to its velocity).

71 1. What is the resultant velocity of an airplane that travels 150. km/h [N] with respect to the if there is a wind blowing 30.0 km/h [E] with to the?

72 B)What heading would the plane need if the pilot wanted to head directly East? This is known as crabbing.

73 Crabbing Videos byyhfgm daxdhm4

74 2. A ship has a velocity of 12.0 km/h West with respect to the water. The water has a current that flows North at 4.0 km/h. Sketch a diagram of the resultant velocity of the ship and calculate the resultant velocity of the ship.

75 3. A boat attempting to travel due East across a 45.0 m wide river at 5.0 m/s encounters a current flowing South at 3.0 m/s. What heading must the boat set if it is to travel due East across the river? Provide a sketch to accompany your calculations.

76 Practice Text: 101 #6

77 Accelerated motion Motion with changing velocity Examples? Accelerated motion occurs when either the speed is changing speeding up or slowing down Direction is changing Curved motion

78 Equations for accelerated motion Formula Sheet??

79 Examples 1. A cyclist can accelerate from 10. km/h to 30. km/h in 3.5s. Find his acceleration.

80 2. A glove is dropped from a chairlift and hits the ground 2 seconds later. A) How high is the chairlift?

81 B) How fast was the glove travelling when it hit the ground?

82 3. Police measure skid marks to determine the speed of a car just before skidding started. If the acceleration of car is 7.5 m/s 2 (based on the surface condition of the road) and the length of the skid is 45 m, determine the speed the car was traveling before locking the tires. (Assuming the car was stopped at the end of the skid.)

83 100 m 4. Determine the speed the object hits the ground. The object has an initial upwards velocity of 25 m/s.

84 5. Determine the time it would take a rock to hit the ground if: A) it was dropped from a height of 2 m. B) thrown upwards at 20 m/s from the 2 m height. C) thrown downwards at 20 m/s from the 2 m height.

85 Free Falling Objects Questions

86

87 What is projectile motion? Anything that is thrown that has some horizontal motion. Jumping on a bike, skis, snowboards, skidoos, skateboards, wakeboards, horses, etc. Running off a diving board. Arrows, or bullets that are shot. Throwing footballs or basketballs, or kicking a soccer ball.

88 A projectile is any object which, once projected, continues its motion by its own inertia and is influenced only by the downward force of gravity.

89 Have you ever used a hose to project the water at a maximum horizontal distance? What did you do to get that maximum distance? What is the best angle at which a projectile can be launched to achieve the maximum range?

90 Why study projectile motion? It is important to study this if you want objects to land in certain spots.

91

92

93 Bad Physics

94 What describes the path of a projectile? Parabola (part or whole) In 3204 projectile motion is limited to negligible air resistance ctile-motion

95 Types of Projectile Motion Horizontal motion moving off a cliff (No vertical motion initially)

96 Types of Projectile Motion Projected at an angle above the horizontal and lands with a net vertical displacement of zero.

97 Types of Projectile Motion Projected at an angle and lands at a point above.

98

99 Projected at an angle (above or below the horizontal) and lands below the launch position and at some horizontal distance (range).

100 Projectile motion problems usually require resolving velocity vectors into its components. Horizontal Motion is uniform motion. (x-direction) no acceleration Vertical Motion is freefall. (y-direction) uniform acceleration (a = -9.8 m/s 2 ) Horizontal Distance is defined as the range. Time is the same in both the x and y direction.

101 Truck and ball

102 Equations for Projectile Motion R Horizontal Motion d (x-direction) Uniform Motion x Range x v t x v t Vertical Motion (y-direction) Accelerated Motion 1 2 d v t at y 1y 2 ad v v y 2 y 1 y d a y v v v 2y 1y t v 2y 1 2 y t

103 Note: The signs in the equations and quadratic formula are essential. Downwards will always be negative (e.g., negative displacement, negative velocities when an object is thrown downwards). You will not be expected to find the launch angle (other than zero) of a projectile.

104 You will be expected to determine the following: range final velocity initial velocity max height, and v X and v y at any point. time of flight Also, either using the quadratic formula or a multi-step process (e.g., finding final velocity first, then time) to solve these problems is acceptable.

105 Components of Velocity Mathematically determine the components of velocity at certain points along a projectile s motion. Applet v 42 m s 35 o

106 A) Find the Components of Velocity of the cannon ball for each of the first five seconds Note: Launch angle is 0 o and initial velocity is 20 m/s B)Find the velocity at each second

107 In the last problem the cannonball was in the air for 5 seconds. A) What is the height of the cliff? B) What was the range of the cannonball?

108 Projectiles with zero vertical velocity 1. A) A Hot Wheels car rolls off a lab bench (h = 1.2 m) at 3.0 m/s. How far would it travel horizontally (range) before hitting the ground?

109 B) If the end of the ramp is raised to a height of 1.00 m, what will be the range of the car?

110 2. A diver runs horizontally off a 3.2 m high cliff. What is the minimum speed required to clear the rocks that extend outward 0.97 m from the bottom of the cliff?

111 Projected at an angle above the horizontal and lands with a net vertical displacement of zero. 1. A soccer ball is kicked into the air with an initial velocity of 50. 0m/s at an angle of 60.0 o from the horizontal. A) Find the initial vertical and horizontal velocity. B)Find the time it takes for the ball to (i) reach its highest distance (ii) return to the earth.

112 Projected at an angle above the horizontal and lands with a net vertical displacement of zero. C) What is the velocity of the ball when it hits the ground? D) How far did the ball travel horizontally?

113 Projected at an angle above the horizontal and lands with a net vertical displacement of zero. 2. A) What is the range of a football that is kicked at an initial speed of 20 m, at an angle of 35 o above the horizontal? B) What is the maximum height of the football?

114 Projected at an angle and lands at a point above. 1. A cannon on a level plain is aimed 42.0 o above the horizontal and a shell is fired with a muzzle velocity of 150. m/s toward a vertical cliff that is m away. How far above the bottom of the cliff does the shell strike the side of the cliff?

115 2. A basketball is released at a velocity of 15 m/s 52 o above the horizontal. The player is at the 3 point line 6.07 m from the basket, and the ball is initially 72.3 cm below the hoop. Will the ball go in?? R

116 Projected at an angle (above or below the horizontal) and lands below the launch position and at some horizontal distance (range). 1. A projectile is thrown from the top of a building which is 26.0 m high. The initial speed of the projectile is 14.0 m/s and it is thrown upwards at an angle of 25.0 o above the horizontal. A) Determine its time in the air.

117 1. A projectile is thrown from the top of a building which is 26.0 m high. The initial speed of the projectile is 14.0 m/s and it is thrown upwards at an angle of 25.0 o above the horizontal. B) Calculate its velocity as it strikes the ground. C) Find its range.

118 2. A cannon that is on a hill 525 m above the ground shoots a cannon ball at 116m/s, 65.0 o below the horizontal. Calculate the range.

119 Some Random Questions 1. A strike in baseball occurs between 0.50 m and 1.0 m directly above home plate. A pitcher, 18 m from home plate, releases a ball, in line with the plate, parallel to the ground. If the ball is released 2.0 m above the ground, with a velocity of 36.0 m/s, would the pitch be a strike?

120 2. A cannon on a level plain is aimed 52.0 o to the horizontal and a shell is fired with a muzzle velocity of m/s toward a vertical cliff. After the shell reaches maximum height, it hits the cliff at a height of 452 m above the plain. What is the horizontal distance between the cannon and the cliff?

121 3. A pumpkin canon fires a pumpkin at speed of 30.0m/s with an angle of elevation of 30.0 o towards a building 30.0m away. The building is 40.0m in width and 10.0m high. Will the pumpkin clear the building? Include a diagram in your solution.

122 Lab Type Question A Hot Wheels TM car is released from a 75.8 cm high tower. It travels 1.5 m down a frictionless track and launches horizontally off a 98.2 cm high table. Find the range of the car.

123 Assignment

124 Projectiles with zero vertical velocity Worksheet 1 Glenbrook applet for answers

125 Projectiles with NON zero vertical velocity Worksheet 2

126 Worksheet 3

127 Assignment

128 The Physics of Juggling

129 Applet Air resistance

130

131

132

133

134

135

136 Page #54. d 1 at 2 2 t 2d a t 2(553 m) 9.80 m/ s 2 The time is 10.6 s.

137 #55. 1 d vt 1 at 2 2

138 #65.

139 #66. Back

140