Topic 2.1: Kinematics. How do we analyze the motion of objects?

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1 Topic.1: Kinematics How do we analyze the motion of objects?

2 Characteristic Graphs The most common kinematics problems involve uniform acceleration from rest These have a characteristic shape for each of s-t, v-t, and a-t graphs

3 Displacement vs. time Displacement (m) Time (s)

4 Information from Graphs We can get information from a graph using one of three strategies: Read data directly from the graph Calculate the slope Calculate the area under the graph

5 Velocity vs. time Velocity (m/s) Time (s)

6 Acceleration vs. time Acceleration (m/s ) Time (s)

7 Graphical Integration If we go from s v a by finding slope, how do we go the opposite direction? Area! The area under the a-t graph gives v, the area under a v-t graph gives displacement s!

8 Acceleration vs. time Acceleration (m/s ) v u at A at Time (s)

9 Ex 1: Find the initial velocity of a car that takes 3.0 s to accelerate at 4.9m/s up to 4 m/s Variables: a=4.9m/s u=? v= 4m/s t= 3.0s v u u v at at u 4m 4.9 m 3s s s 7 m s

10 Displacement If we got velocity from an a-t graph, can we get displacement from a v-t graph?

11 Velocity vs. time Velocity (m/s) u s ut 1 at A 1 bh 1 t A lw ut at Time (s)

12 Ex 1: Find the displacement of a car that accelerates for.5s from a stop light at 3.5 m/s Variables: a=3.5m/s u= 0 s=? t=.5s s s 1 ut 3.5m s s 11m 1 at. s 5

13 Ex : Find your reaction time, given the ruler is falling at -9.8 m/s Variables: a=-9.8m/s u= 0 s= -0.11m+/-0.01 t=?s s t t ut 1 at s a ( 0.11) 9.8 t 0. 15s

14 Practice: Read p Watch Ringo Exercises p. 53 #1-8

16 How does gravity work? Do we disappear if we sail off the edge of the Earth?

17 Free Fallin

18 Gravitational Acceleration g How fast do things fall? First, do heavy objects fall faster than light objects? Try it!! If all free falling objects fall with the same acceleration, can we find this elusive g? Throw a golf ball, take a video Graph the variation of vertical velocity with time, find the slope!!

19 Gravitational Acceleration g How fast do things fall? First, do heavy objects fall faster than light objects? Try it!! If all free falling objects fall with the same acceleration, can we find this elusive g? Drop a ball from a height of 1.00 m time its fall, solve for g!! s 1 gt ut s 1 gt s g g m t? s

20 Ex 1: How long should a ball take to fall 1.00m? Variables: a=g u= 0 s= -1.00m t=? t s ut 1 at t 0. 45s 1.00m 9.8m s t s g

21 Showing your work t 1.00m 9.8m s Without the why you are powerless!

22 Ex : Find the position of a ball that is thrown up at 4 m/s, after 4.5s of flight Variables: a=g u=4m/s s=? t= 4.5s s s 1 at ut 1 4m 4.5s 9.8m 4. 5s s s 8. 8m s

23 Projectiles Any object in free fall

24 Yes, even you!

26 What do we know about free fall? Horizontal acceleration is zero (neglecting air resistance) Vertical acceleration is -9.8m/s (on Earth) The path is always a parabola We can solve each component independently, using t to connect them

27 Variables: Ex 1: A student runs off a 10m cliff at 8.5 m/s a) How far from the bottom does he land? a=-9.8m s - s=-10m u= 8.5 m/s t=? x=? x 1 u x u cos 8.5m s u y u sin 0 1 s u t 1 at x ut at y x 8.5m s s t y x 1m s a ( 10m) t 1. 43s 9.8m / s

28 b) With what velocity does he hit the water? x y u u m x cos 8. 5 s u u sin 0 y v y u y gt v y 9.8m / s1.43s 14m / s v v x v y v 16m / s tan 1 v v y x 59

29 Exercises Continue p. 53 #1-33 FIAQD

30 Special Cases Horizontal Launch: u y =0 Vertical launch: u x =0 Max height: v y =0 Level ground: v y =-u y range formula:r=(-u sin)/g Max. range for 45

31 Ex 1: Find the location at maximum height for a projectile launched at 3m/s, 41 Variables: g=-9.8m/s u= 3 m/s Ө=41 y=? t=? x=? u cos 4 m s s ut s 4(.14) s 51m x s s y t u sin 1m / v v u a u v 1 0 u t t at s m s

32 Ex : Golf ball launched at 48m/s, 45 above horizontal. Find range for a level Variables: g=-9.8m/s v 0 = 48 m/s Ө= 45 t=? x=? u x u cos s ut landing x 33.9m / s 1 at s d 35m y u y u sin 33.9m/ s s ut 1 at 1 t( u at) t u / a t 6. 9s 0

33 Ex : Motorcycle launched at 35m/s, 35 above horizontal. Find range for a level Variables: g=-9.8m/s v 0 = 35 m/s Ө= 35 t=? x=? landing x u u m x cos 9 s s ut s 9m / s 4. 1s s 118m y v u y u sin y t vy 0 v g t 4. 1s gt 40m / s 9.8m / s

34 Ex 3: A student launches her textbook out a 5m window at 3m/s, 15 above horiz. Find x Variables: g=-9.8m/s v 0 = 3 m/s Ө= 15 y=-5m t=? x=? u cos.m s s ut 1 x.m / s 1. 78s x 40m v v v y u sin 5.95m s v u as u m s at 1 ( 9.8)( 5) 1 t t 1. 78s v u a

35 How can you control your flight?

36 Ex 4: Projectile launched at 3m/s, 41 up onto a 6.5m roof. Where does it land? Variables: g=-9.8m/s v 0 = 3 m/s Ө=41 y=6.5m t=? x=? x v v m x 0 0 cos 4 s v y 0 v0 sin 1m / s x v t 1 y vy t gt x 4m / s3. 95s x x 95m y 4.9t x t b 0 1t b 4ac a s 9.8 t 0. 34s t 3. 95s

37 Puzzler! Projectile launched at 17m/s, 3 up onto a 8.0m roof. Where does it land? Variables: g=-9.8m/s v 0 = 17 m/s Ө= 3 y=8.0m t=? x=? x y v v m x 0 0 cos s v y 0 v0 sin 9.0m / s x v t 1 y vy t gt x 4.9t 0 9.0t b b 4ac x a 18.7 t s

38 Real Life Ex 1: Projectile launched at 15 above horizontal. Find the launch velocity Variables: g=-9.8m/s v 0 =? Ө= 15 t= x=6.08m x v x v t v x 0 v 0 cos v 0 x x t v x 0 cos 6.08m s x cos15 y 8.03m / s

39 Real Life Ex : Projectile launched at 8.66m/s, theta above horizontal. Find Variables: g=-9.8m/s v 0 =9.01 m/s Ө= choose t= x=? y=-1.18m range and compare (% diff) v x 0 v 0 cos x v x t x y v y 0 v 0 sin y v y t 4.9t 0 1 gt.87t b b 4ac t 0.34 s,0.6813s a

40 Real Life Ex 1: Projectile launched at?m/s, 35 above horizontal. Find range and compare (% diff) The x=7.50 Exp x=7.80m % diff Experimental Theoretical Theoretical % diff 7.80m 7.50m 7.50m % diff 4.0%

41 Real Life Ex 3: Projectile launched at 13m/s, 45 above horizontal. Find range Variables: g=-9.8m/s v 0 = 13 m/s Ө= 45 t= x=? v x 0 v 0 cos 13m v x 0 s v x 19 m 0 9. x v x 10. m s s x t x 7. 4m cos45 s x t 0 y v y 0 v 0 sin 13m v y 0 s v y 19 m 0 9. y v y t 0 4.9t b b 4ac a sin45 1 gt s 9.19t s, s

42 Real Life Ex 4: Projectile launched at 13 m/s, 3 above horizontal. Where does the Variables: a=-9.8m/s u= 13 m/s Ө= 3 t=? X=? y=1.m u x u cos s ut ball hit? x vx0 13m cos 3 s v x m s s 11.97(0.66) s 7. 9m y u y u sin s ut 4.9t 1 at 5.08t t 0. 66s m s 1. 0

43 Real Life Ex : Projectile launched at?m/s, 35 above horizontal. Find range and compare (% diff) The x=9.48 Exp x=9.50m % diff Experimental Theoretical Theoretical % diff 9.50m 9.51m 9.51m % diff 0.1%

44 Real Life Ex 3: Projectile launched at 9.7m/s, 15 above horizontal. Find range Variables: g=-9.8m/s v 0 = 9.7 m/s Ө= 15 y=-1.15m t=? x=? and compare (% diff) x y v v m x 0 0 cos s v y 0 v0 sin.40m / s x v t 1 y vy t gt x 8.95m / s0. 79s x x 7. 05m 4.9t x b 0.40t t 0. 79s b 4ac a t s t 0. 30s

45 Real Life Ex 3: Projectile launched at 9.7m/s, 15 above horizontal. Find range and compare (% diff) The x=9.05 Exp x=9.08m % diff Experimental Theoretical Theoretical % diff 9.08m 9.05m 9.05m % diff 0.3%

46 Experiment Design and carry out a lab, measuring launch velocity and angle Calculate horizontal displacement, then compare to measured displacement

47 % difference The x=9.05 Exp x=9.08m % diff Experimental Theoretical Theoretical % diff 9.08m 9.05m 9.05m % diff 0.3%

48 Exercises Start p #5-33

49 One does not simply mix up their x and y variables

51 What if we don t know time? We can combine these two equations to eliminate t: a u v t 1 at ut s at u v 1 a u v a a u v u s 1 a u vu v a a u uv s u vu v u uv as

52 v u as Variables: a=? u= 35m/s v= 0 s=59m Ex 3: Find the acceleration of a car that stops from 35 m/s in a distance of 59 m a v u as m 0 35 *59m s v u s m a 10.4 s a

53 The last equation of this unit solves problems involving quadratics: 4 b b ac x a Ex 4: How long does it take for a ball thrown upwards at 13 m/s to reach a height of 3. m? 1 s ut at

54 Ex 4: How long does it take for a ball thrown upwards at 13 m/s to reach a height of 3. m? Variables: a=g u= 13m/s t=? s=3.m b x s ut 4.9 m b 4ac a 1 at 1 at ut t 13m t 3.m s s t s t s t. 4s t 0. 8s

55 Ex : When does a ball thrown upwards from a 4.5m roof at 3. m/s hit the ground? Variables: a=g=- 9.8m/s u= 3.m/s t=? s=-4.5m b x s ut 4.9 m b 4ac a 1 at 1 at ut t 3. m t 4.5m s s t s t s t 1. 34s t 0. 69s

56 Ex 3: When does a ball thrown upwards at 1.4 m/s hit the 3.m high ceiling? Variables: a=g v 0 = 1.4m/s t=? d=3.m b x s b 4ac a ut 4.9 m t 1 at 1 at ut t 1.4 m t 3.m s s s t 1.4 err? 9.8 s

57 Which one? A quadratic yields two roots. Sometimes the physics rules out the possibility of one of the roots eg. negative time, etc. In the first case, we had two solutions to the problem: one on the way up, one on the way down

58 Height (m) time (s)

59 Ex 4: How long does it take for a ball thrown upwards at 13 m/s to reach a height of 3. m? Solve without q.f? Variables: a=g u= 13m/s t=? s=3.m v u as v u as v v 10.3m s 1

60 Ex 4: How long does it take for a ball thrown upwards at 13 m/s to reach a height of 3. m? Solve without q.f? Variables: a=g u= 13m/s t=? s=3.m v u at v u t a t 9.8 t 0. 8s

61 Problem solving strategy Step 1: start with what you know Variables: a= u= t=? s= Step : find a formula containing only those variables v u Step 3: rearrange if necessary at Step 4: insert numbers and calculate Step 5: bask in the admiration of all your family and friends :) t v a u

62 Practice: Continue Chapter Review Problems p #11-17 Finished? Start Test Yourself p. 43 #1-8

63 Scalars vs. Vectors Scalars: have a magnitude but no direction: mass time distance speed Vectors: have magnitude and direction force momentum displacement velocity

64 Adding Vectors We always rearrange vectors to add them tip-to-tail. The resultant is the vector that reaches from the tail of the 1 st to the tip of the nd

65 Adding Vectors Ex: Draw a scale diagram for: 5 m North + 3 m South 4 N East + 3 N South 10 m North + 5 m Northeast

66 Subtracting Vectors We can think of vector A B as being equivalent to A + (-B) A A B B A B A B

67 Exercises P. 18 #1- P. 19 #1

68 Graphs Must have: title appropriate labels, scaled axes best fit line, not connect-the-dots! Should take up most of the page Should give us useful information

69 Displacement Distance is a measurement of how far you have travelled and depends on the path d 57km 1 d Displacement is independent of path because it simply measures the change in position km West

70 Ex 1. Eric walks 15 blocks North then 10 blocks South. Find his: distance: d=15 blocks + 10 blocks=5 blocks displacement d 15blocks ( 10blocks) N d 5blocks North Ex. Ken walks 7 m East then 10 m North. Find his displacement :

71 Ex. Ken walks 7 m East then 10 m North. Find his displacement : c b a 7 10 d d 1m 7 10 tan tan 1 1 x y NofE m d 55, 1

72 Components What is the opposite of adding perpendicular values to get a D vector? Finding Components Ex: Vector A=55N, 30 above horizontal A x Acos 48N A Asin y 8N

73 Ex : Motorcycle launched at 35m/s, 35 above horizontal. Find range for a level Variables: a=-9.8m/s u= 35 m/s Ө= 35 t=? x=? landing x u u m H cos 9 s s u H s 9m / s 4. 1s t v V y u u m V sin 0 s uv at s uv t 0 u V t 1 at 0 t u 1 V at v 40m / s t 4. 1s g 9.8m / s x 118m 1 at

74 Ex 3: A student launches her textbook out a 5m window at 3m/s, 15 above horiz. Find x Variables: g=-9.8m/s v 0 = 3 m/s Ө= 15 y=-5m t=? x=? v x vx0 v0 cos. m s v y 0 v0 sin 5.95m / s x v t 1 y vy t gt x x.m / s*1. 78s x x 40m y 4.9t x b t t 1. 78s b 4ac a 5 0 t 0. 57s

75 Exercises Finish experiment p. 53 choose 10

76 Time for a trip to the library?

77 Free Body Diagrams The point of a FBD is to simplify the dynamics involved We only point out the forces acting on the body in question To get to the point, we draw the body as a point! F 1 F The forces are drawn pointing away from the body F 3

78 How do you reach equilibrium?

79 Translational Equilibrium This is a fancy way of saying a=0 It then follows from Newton s nd that: or F 0 F 0 net

80 Since we deal with Newton s nd law in one dimension at a time, we can also say: and x y F 0 F 0 Ex 1: Traffic light equilibrium find the weight of the traffic light from the following diagram:

82 Ex: draw a FBD for a book resting on your table R F 0 F g If the book weighs 13 N, find R Fg+R=0 R=-Fg=-(-13N)=13N

83 Ex: draw a FBD for a book sliding across your table R F f F g

84 Ex: draw a FBD for a book being pushed with constant velocity across your table R F f F a F g

85 Ex: draw a FBD for a rocket ship in space F t

86 Draw a free body diagram for this traffic light F 1 F F g

87 Find F if F1 is 47N of tension F F F g

88 Find F if F1 is 47N of tension Equilibrium, so Fg+F1+F=0 F=47/tan5 F=334N 38 F 5 F F g

89 Read p P #17-1 Exercises

90 What if we all jumped at once?

91 Newton s 1st Law Objects with mass have Inertia: the tendency to stay at rest (or moving!) The more mass an object has, the more difficult it is to move it (or stop it!)

93 Percent Difference Experimental Theoretical % Diff 100% Theoretical % Diff 100% %

94 Evaluation Your conclusion should describe the results, relevant to the research question (forces and equilibrium), and supported by the data List limitations and sources of error. Address each with suggestions for improvement

95 Mini Quiz: 1) Find T (3 marks) ) What will happen to the magnitude of T1 if we increase from 4 to a larger angle? (3 marks) T1 4 T 15N

96 Mini Quiz: Find T sin104 sin 48 F 47N g T1 T 15N

97 MOMENTUM IS THE PRODUCT OF MASS AND VELOCITY p=m v

98 What is the momentum of a 10 kg rugby player running at 11 m/s? p=mv p=10kg 11 m s -1 /s p=130 kg m s -1

99 Newton s nd Law Impulse": change in momentum Impulse Ft p

100 Can we find the impulse Venus applies to this 57g tennis ball? She returns a 46m/s serve at 35m/s Impulse p mv mu 0.057(35) 0.057( 46) 4.6kg m s 1

101 p. 38 #4-5 Start Personal Engagement assignment Using the ipads, write an introductory paragraph for a possible IA topic. Include a research question, and reasons why you chose that particular topic

102 Can we find the Force Venus applies to this 57g tennis ball? She returns a 46m/s serve at 35m/s and the ball only contacts the racquet for 6.5ms (35) ( 46) Ft p F p t

103 Newton s nd Law Rearranging the impulse formula gives us: F ma Ex 1: how much force is necessary to accelerate a 80kg student at 10 m/s? F ma 80(10) 800N

104 What do we mean net force? Net force is zero if there are no unbalanced forces We usually do not notice forces until they become unbalanced Ex. What are the forces acting on a suction cup?

Find the acceleration of this 300 kg rocket T 4500N F mg 3009.

105 Find the acceleration of this 300 kg rocket T 4500N F mg a F m a 4500N ( 940N 300kg ) a 5.m s 5.m 9.8m s s 0.53g W 940N

106 Find: A) the acceleration of these masses, neglecting friction W mg 69.8 a F m a 58.8N (8 6) kg a 4.m s 58.8N

107 Find: b) the tension in the string F ma T m s T 33. 6N

108 Find R for this 55 kg elevator passenger F ma W R ma R W ma R ( 539) 55( 3.) 3.m s R 363N

109 p. 41 #6-31 Start Personal Engagement assignment Using the ipads, write an introductory paragraph for a possible IA topic. Include a research question, and reasons why you chose that particular topic

110 Newton s 3 rd Law For every action there is an equal and opposite reaction When you hit something, it hits back!

111

Normal Force When an object is in contact with a

112 Normal Force When an object is in contact with a supporting surface, it pushes down on that surface Newton s 3rd Law states the surface pushes back with an equal and opposite force This is often (but not always!) equal to the object s weight We sometimes refer to Normal force as the apparent weight

113 Simple case: object at rest Ex 3: What is the normal force acting on the.5 kg book resting on your desk? What forces act on the book? Gravity and Normal force Free body diagram Apply nd law F N F g Fnet ma 0

114 F g F N 0 F F mg N g F N F.5kg 9.8 N 4.5N N kg F g

115 Extended object at rest Ex 3: what is the normal force acting on your book as you lean on it with a 35 N force? What forces act on the book? Gravity, Applied and Normal force Free body diagram Apply nd law F g F N F a Fnet ma 0

116 Fnet ma 0 F F F g a N 0 F N F F F N g a F ( 4.5 N) ( 35 N) N F g F a FN 60N

117 Accelerating object Ex 4: find the apparent weight of a 50 kg student accelerating upwards at 3.4 m/s What forces act on the student? Gravity and Normal force Free body diagram Apply nd law F N F g Fnet ma

118 F N F g ma F N F g ma F N mg ma F N F N F 660N N F g

119 p. 41 #3

120 GRF Mini Quiz Use the graph from the force platform to a) identify regions of positive, negative and zero acceleration (4 marks) b) Solve for the maximum positive acceleration (3 marks) c) What differences would you see on the graph if this hefty fellow gained even more mass over the holidays? (3 marks)

121 Principle of Momentum Conservation: Total momentum before is equal to total momentum after the collision This is true for all closed, isolated systems No one allowed in or out No net external force

122 Chabal: timing is everything

123 Can we use momentum to analyze this collision?

124 Momentum is conserved so p before must equal p after Rokocoko has a mass of 105 kg and runs at 7.5 m/s into 9 kg Betson How fast are they moving after the collision?

125

126 Most common physics 11 collision: moving object collides with stationary one; they move off together Ex: a student with a mass of 105 kg runs at 7.5 m/s into a 9 kg classmate, find v

127 Same steps every time! Variables m 1 = u 1 = m = u = v'= p i =p f p 1+ p =p 1 ' + p ' m 1 u 1 + m u = m 1 v 1 + m v m 1 u 1 + m u = (m 1 + m )v? 0 = m 1 v 1 + m v?

128 For an explosion? Ex: bottle rocket Variables m 1 =0.30 kg v 1 =? m =0.850 kg v =-5 m/s p i =p f 0 = m 1 v 1 + m v m 1 v 1 =- m v v 1 =- m v /m 1 v 1 =-0.850kg(-5m/s)/0.30kg v 1 = 66 m/s

129 Try p. 46 #33, 34

130 Area under the graph This should give impulse

131 Ex: a) Find the impulse b) Find the change in velocity of the 0.3 kg ball

132 Finish p. 46 #33, 34

Chapter 4: Dynamics. Newton s Laws

Chapter 4: Dynamics. Newton s Laws Chapter 4: Dynamics Newton s Laws What if we all jumped at once? Newton s 1st Law Objects with mass have Inertia: the tendency to stay at rest (or moving!) The more mass an object has, the more difficult