Chapter 2: 1-D Kinematics

Transcription

1 Chapter : 1-D Kinematics

2 Types of Motion Translational Motion Circular Motion Projectile Motion Rotational Motion

3 Natural Motion Objects have a proper place Objects seek their natural place External forces must be constantly applied to moving objects in order to keep them going. The heavier the object, the faster it falls. Did not experiment to test theories.

4 Galileo Challenged The Dogma Of Natural Motion with Experiments

5 Galileo Challenged The Dogma Of Natural Motion The natural motion of a body is to remain in whatever state of motion it is in unless acted upon by net external forces.

6 Galileo Challenged Aristotle Physics In a vacuum, all objects fall with the same acceleration due to gravity: 9.80 m/s, independent of their weight.

7 Galileo s Motion Studies gave us Definitions: x v t v0 v v v a t f

8 Distance and Displacement The total distance traveled relative to an origin. Distance is a scalar. Displacement is a vector. The unit is the meter. x x x f 0 (delta) means "change in" = 'final - initial'

9 QuickCheck.1 An ant zig-zags back and forth on a picnic table as shown. The ant s distance traveled and displacement are A. 50 cm and 50 cm. B. 30 cm and 50 cm. C. 50 cm and 30 cm. D. 50 cm and 50 cm. E. 60 cm and 30 cm. Slide -9

10 QuickCheck.1 An ant zig-zags back and forth on a picnic table as shown. The ant s distance traveled and displacement are A. 50 cm and 50 cm. B. 30 cm and 50 cm. C. 50 cm and 30 cm. D. 50 cm and 50 cm. E. 60 cm and 30 cm. Slide -30

11 Average Speed &Velocity Speed is how fast something moves. The average speed is the total distance per time. The average velocity is the the total displacement per time. Velocity is a vector. The unit is m/s. v total displacement total time x t

12 Sense of Speed 1 m/ s.5 mi / hr 1 m / s 3.6 km / hr.4 mi / hr 10 m / s 36 km / hr.4 mi / hr 0 m / s 7 km / hr 44.8 mi / hr 30 m / s 108 km / hr 67. mi / hr 1 km / hr 0.6 mi / hr

13 Acceleration How fast How fast is changing. The rate at which the speed is changing. a change in velocity change in time v t Speeding up Slowing down Constant speed, changing direction.

14 Acceleration is in the direction of the net Force but not necessarily in the direction of velocity. Velocity is always in the direction of the motion!

15 Quicky Question An automobile enters a freeway on-ramp at 15.0m/s and accelerates uniformly up to 5.0 m/s in a time of 10.0s. a) What is the automobile s average velocity? Which equation? v v0 v f 15 m / s 5 m / s 0 m/ s

16 Quicky Question An automobile enters a freeway on-ramp at 15.0m/s and accelerates uniformly up to 5.0 m/s in a time of 10.0s. b) What is the automobile s average acceleration? Which equation? a v t v f t v i 5 m / s 15 m / s 10s 1 m/ s

17 Quicky Question An automobile enters a freeway on-ramp at 15.0m/s and accelerates uniformly up to 5.0 m/s in a time of 10.0s. c) What is the distance traveled in this amount of time? Which equation? x v x v t (0 m / s)(10 s) t x 00m

18 Motion Diagrams

19 Draw the Motion Diagram An automobile enters a freeway on-ramp at 15.0m/s and accelerates uniformly up to 5.0 m/s in a time of 10.0s.

20 Skiing through the woods

21 QuickCheck.13 Here is a motion diagram of a car speeding up on a straight road: The sign of the acceleration a x is A. Positive. B. Negative. C. Zero. Slide -69

22 QuickCheck.13 Here is a motion diagram of a car speeding up on a straight road: The sign of the acceleration a x is A. Positive. B. Negative. C. Zero. Speeding up means v x and a x have the same sign. Slide -70

23 Which position-versus-time graph represents the motion shown in the motion diagram?

24 Which position-versus-time graph represents the motion shown in the motion diagram?

25 Galileo s Motion Studies gave us Kinematic Equations x v v f t 0 v, v, a v t With a little al-jbr.

26 Start: x v v f t 0 v, v, a v t Assume constant acceleration! v a t v v a t f 0 v v a t f 0

27 Start: x v v f t 0 v, v, a v t Assume constant acceleration! v a t v v a t f 0 v v a t f 0

28 Start: v0 v f x v, v, a t v t x t v0 v f xf xi v0 ( v0 a t) t 1 x f xi v0 t a t

29 Start: v0 v f x v, v, a t v t v0 v f x t t v0 a v t t v = f v v a f 0 x Combine & Eliminate t: t v v f 0 x v v a 0 f Algebra: f 0 v v a x

30 Galileo s Motion Studies gave us Kinematic Equations x v v t 0 f v, v, a v t v v at f 0 1 x v0t at v v a x f 0

31 Problem Solving Strategy

32 Acceleration: Changing Velocity From t = 0, how long does it take the car to come to a full stop? How far does the car travel before it comes to a stop? Draw the Motion Graph +x

33 Acceleration: Changing Velocity Knowns a v v i f 5 m / s 8 m / s 0 t? Which equation to use? f Solve for t: v v at i v t f a v 0 8 m/ s 5 m/ s i t 5.6s 5.6s

34 Acceleration: Changing Velocity From t = 0, to t = 5.6s, how far does the car travel before it comes to a stop? Knowns Which equation? 1 a 5 m / s x v t at v v i f 8 m / s 0 t 5.6s m 1 x s ( 5 m / s )(5.6 s) 78.4m s 0 YOU TRY IT! x 78.4m +x

35 Brake Question You are driving a car going 80 km/hr (50mph) when a head-on collision happens 5 meters ahead of you. If you can brake at 6 m/s, will you hit the crash or stop before it? Knowns : v 80 km / hr m/ s, v 0, a 6 m/ s Unknown : x? 0 f 0 v v a x ( 6 m/ s ) x 0 v a ( m/ s) x 40.3m 5m f Stopping Distance goes as the SQUARE of the speed! CRASH!

36 Stopping Distance Traveling at 70 miles per hour, what is your breaking distance? v v a x f 0 x 0 v a If v doubles, d quadruples!!! Stopping Distance goes as the SQUARE of the speed!

37 Motion Graphs What kind of motion is this?

38 What kind of motion is this?

39 What is the velocity during each segment? v x t v 0 m/ s v 3 400m 400s 1 m / s v 1 400m 00s m / s

40 Slide -114

41 Galileo s Motion Studies gave us Definitions of averages x v v t 0 f v, v, a v t Kinematic Equations with constant acceleration v v at f 0 1 x v0t at v v a x f 0

42 Constant vs Changing Acceleration Depends on the FORCE Constant Forces Constant pushes and pulls Inclined planes Gravity near the earth (Free Fall) Pulleys, Conical Pendulums Variable Forces Springs and Pulleys Air Resistance Gravity Far from Earth Electricity and Magnetism MOST FORCES!!!!

43 Free Fall Unless told otherwise, ignore air resistance for free fall problems!

44 Galileo Challenged Aristotle Physics In a vacuum, all objects fall with the same acceleration due to gravity: 9.80 m/s, independent of their weight.

45 Acceleration of Freely Falling Object The acceleration of an object in free fall is directed downward, regardless of the initial motion The magnitude of free fall acceleration is g = 9.80 m/s g decreases with increasing altitude g varies with latitude 9.80 m/s is the average at the Earth s surface We will neglect air resistance g is a SCALAR!!! POSTIVE

46 Free Fall Equations For any object in the absence of air resistance. a g 9.80 m/ s y (taking up as +y) Kinematic Eqs: v v at f 0 1 x v t at 0 0 v v a x f Customize: v v gt f 0 1 y v0t gt v v g y f 0 Note: v 0 can be negative!

47 Falling from Rest v0 0 Estimate : v v gt f a g ~ 10 m / s 0 1 y v0t gt Take down as +y: + v 10 m / s y 5m v 0 m / s y 0m v at 10t 1 y at 5t How FAR is not How FAST! v y! v 30 m / s y 45m v 40 m / s y 80m v 50 m / s y 15m

48 How Far: y(t) ~ t 1 y v t at 0 + How Fast: v(t) ~ t 1 v v at How Fast How Fast is f Changing! g m / s

49 FIRST: Define Reference Frame 9.80 m/ s In this reference frame,what is the sign of a? What is v at t = 3s? Knowns : v0 0, a 9.8 m/ s, t 3s Unknown : v f? a v v at f 0 m (3 s ) s 9.4 m s Negative because it is moving downward, in the negative direction!

50 FIRST: Define Reference Frame How far did the ball fall in those 3 seconds? Knowns v a m s t s v m s Unknown : y? : 0 0, 9.8 /, 3, f 9.4 / 0 1 y v t at 1 m 0 ( 9.8 )(3 ) s s 44.1m The ball fell 44.1m. The displacement is negative because it is moves downward, in the negative direction but how far is a distance a scalar and is positive!

51 Throwing up is Also Free Fall! Symmetry of G Field. Estimate : a g ~ 10 m / s v v gt f 0 1 y v0t gt

52 What Goes Up Must Come Down Someone standing at the edge of a cliff throws one ball straight up and one straight down at the same speed. Ignoring air resistance, which ball strikes the ground with the greatest speed?

53 Free Fall Question: You throw the rock down with an initial speed of 30 m/s. The rock hits the ground in 3 seconds. With what speed will the rock hit the ground? Knowns v m s a m s t s Unknown : v f? v v at f 0 : 0 30 /, 9.8 /, 3 +y v f m m (3 ) s s s m 59.4 s How high is the cliff?

54 Free Fall Question: You throw the rock down with an initial speed of 30 m/s. The rock hits the ground in 3 seconds. With what speed will the rock hit the ground? How high is the cliff? Knowns v m s a m s t s Unknown : y? 0 1 y v t at : 0 30 /, 9.8 /, 3 ( 30 m / s)(3 s) ( 9.8 m/ s )(3 s) 134m 1 +y The cliff is 134 m high.

55 Free Fall: Throwing Up What is the speed at the top of the path? ZERO! +y What is the acceleration at the top? a = m/s What is the velocity at the same height on the way down? -30 m/s With what velocity will the rock hit the ground? m/s SAME as if you threw it straight down at 30m/s!

56 Free Fall: Throwing Up Problem How long does it take to hit the ground? First try to guess! How long to the top? How long back to launch point? Final v increases by 30m/s? Knowns v m s a m s t s v m s Unknown : t? : 0 30 /, 9.8 /, 3, f 59.4 / v v at f I guess about 9 seconds! 0 +y t v v 59.4 m / s 30 m / s f 0 a 9.8 m / s t 9.1s

57 QuickCheck.18 A ball is tossed straight up in the air. At its very highest point, the ball s instantaneous acceleration a y is A. Positive. B. Negative. C. Zero. Slide -96

58 QuickCheck.18 A ball is tossed straight up in the air. At its very highest point, the ball s instantaneous acceleration a y is A. Positive. B. Negative. C. Zero. Slide -97

59 Motion on an Inclined Plane Figure (a) shows the motion diagram of an object sliding down a straight, frictionless inclined plane. Figure (b) shows the the free-fall acceleration the object would have if the incline suddenly vanished. This vector can be broken into two pieces: and. The surface somehow blocks, so the one-dimensional acceleration along the incline is The correct sign depends on the direction the ramp is tilted. 013 Pearson Education, Inc. Slide -10

60 The ball rolls up the ramp, then back down. Which is the correct acceleration graph? 013 Pearson Education, Inc.

61 The ball rolls up the ramp, then back down. Which is the correct acceleration graph? 013 Pearson Education, Inc.

62 QuickCheck.5 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? 013 Pearson Education, Inc. Slide -44

63 QuickCheck.5 Here is a motion diagram of a car moving along a straight road: Which velocity-versus-time graph matches this motion diagram? 013 Pearson Education, Inc. Slide -45

64 Rank in order, from largest to smallest, the accelerations a A a C at points A C. A) a A > a B > a C B) a A > a C > a B C) a B > a A > a C D) a C > a A > a B E) a C > a B > a A 013 Pearson Education, Inc.

65 Rank in order, from largest to smallest, the accelerations a A a C at points A C. A) a A > a B > a C B) a A > a C > a B C) a B > a A > a C D) a C > a A > a B E) a C > a B > a A 013 Pearson Education, Inc.

66 QuickCheck.7 Here is a position graph of an object: At t = 3.0 s, the object s velocity is A. 40 m/s. B. 0 m/s. C. 10 m/s. D. 10 m/s. E. None of the above. 013 Pearson Education, Inc. Slide -50

67 QuickCheck.7 Here is a position graph of an object: At t = 3.0 s, the object s velocity is A. 40 m/s. B. 0 m/s. C. 10 m/s. D. 10 m/s. E. None of the above. 013 Pearson Education, Inc. Slide -51

68 Which velocity-versus-time graph or graphs goes with this acceleration-versustime graph? The particle is initially moving to the right and eventually to the left. 013 Pearson Education, Inc.

69 Which velocity-versus-time graph or graphs goes with this acceleration-versustime graph? The particle is initially moving to the right and eventually to the left. 013 Pearson Education, Inc.

70 Which velocity-versus-time graph goes with the position-versus-time graph on the left?

71 Which velocity-versus-time graph goes with the position-versus-time graph on the left?

72 Which position-versus-time graph goes with the velocity-versus-time graph at the top? The particle s position at t i = 0 s is x i = 10 m.

73 Which position-versus-time graph goes with the velocity-versus-time graph at the top? The particle s position at t i = 0 s is x i = 10 m.

74 Speedy Sally Speedy Sally, driving at 30.0 m/s, enters a one-lane tunnel. She then observes a slow-moving van 155 m ahead traveling at 5.00 m/s. Sue applies her brakes but can accelerate only at.00 m/s because the road is wet. Will there be a collision? If yes, determine how far into the tunnel and at what time the collision occurs. If no, determine the distance of closest approach between Sally's car and the van. Sketch the x-t graphs for both the vehicles. What does it mean?

75 Rock Drop A rock is dropped from rest into a well. The sound of the splash is heard 3.0 s after the rock is released from rest. How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is 336 m/s.