Introduction to Kinematics. Motion, Forces and Energy

Transcription

1 Introduction to Kinematics Motion, Forces and Energy

2 Mechanics: The study of motion Kinematics The description of how things move 1-D and 2-D motion Dynamics The study of the forces that cause motion Newton s Laws Energy Momentum

3 Remember: Before we can study something, scientists need to define what it is they are studying. They determine what belongs in the study boundaries the system all processes, forces, and measurable properties. Anything outside the system s boundaries the surroundings. The surroundings can affect the system and these interactions are studied as well.

4 Remember: problem solving method 1. Sketch the problem 2. Choose the formula(s) 3. Calculate your Answer 3. Check it!

5 Kinematics CH 4 Describing motion

6 What is Motion? any physical movement or change in position or place, relative to a reference point Movement Referenceo int

7 understanding vocabulary A stationary landmark or location from which we start or make our observations. Reference point

8 Choose the point of reference or reference frame that is easiest for you to use in solving the problem or answering the question. Make all measurements from that point of reference or reference frame.

9 understanding vocabulary An unchanging rate of motion. Moving at uniform speed.

10 understanding vocabulary The rate of change of speed during motion.

11 Choose the coordinates Establish where 0 is (the origin) Establish directions where the values increase

12 understanding vocabulary the origin (value of 0) is placed at the reference point. Imaginary line marked off in measured units

13 Using a coordinate system example - football What origins and direction are used to determine 1 st down? Length of punt?

14 Motion diagrams A series of images of a moving object that records its position after equal time intervals. At rest Moving at a constant speed It represents the position, velocity and acceleration of an object Speeding at several up different times. Slowing down

15 the particle model

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17 What is the motion of the cart in this diagram? Change is position is less per each time interval time in seconds Slowing down (deceleration)

18 We can use motion diagrams to represent 4 concepts in kinematics: At rest Uniform motion (constant speed) Speeding up (acceleration) Slowing down (deceleration)

19 Demonstrate uniform and accelerated motion Design it, show it, use evidence to support it.

20 Measuring motion in 1 and 2 dimensions SCALARS AND VECTORS

21 understanding vocabulary has magnitude A measurable quantity. Can be positive, negative or zero.

22 understanding vocabulary has magnitude AND direction A measurable quantity that also includes direction. Can be positive or negative.

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24 Position vector Proportional to the distance of the object from the origin and points from the origin to the location of the object at a particular time. + What is smiley s position vector? + 3 m meters +

25 try some. Scalar or Vector? a. 5 m See Answer scalar b. 30 m/sec, East See Answer vector c. 5 mi., North See Answer vector d. 20 degrees Celsius See Answer scalar e. 256 bytes See Answer scalar f Calories scalar

26 Describe the position of the house relative to the bicycle using only scalars. 4 km Describe the position of the house relative to the bicycle using vectors. 4 km, N

27 Is speed a scalar or vector? 1. What is the bike s speed after 1 hour? 25 mi/hr B C 2. Does the bike speed up or slow down at B? How do you know? Speed up, slope increases 3. What is the speed at B? 50 mi/hr 4. What is the speed at C? 0, it is stopped

28 Can you control your speed? Another design challenge example

29 Distance and Direction How far overall?

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31 understanding vocabulary initial time - t i final time - t f t f - t i represents the time interval use the formula: Δt = t f - t i The length of a time span.

32 remember Motion occurs when an object s position changes during an interval of time. One dimensional motion can be represented by position along a single coordinate axis.

33 distance vs displacement Distance is a scalar quantity. It is always positive. It refers to how much ground an object has covered during its motion. There is no direction associated with it.

34 Distance is the length of the actual path taken by an object. Consider travel from point A to point B in diagram below: A d = 20 m B Distance d is a scalar quantity: Contains magnitude only and consists of a number and a unit. ex: 20 m, 40 mi/h, 10 gal

35 Distance = how far an object has moved. Measured in meters, kilometers (cm or mm) Y Axis X Axis If each mark represents 10 cm, what is the distance between the girl and the ball?

36 Displacement is the object's change in position. displacement distance

37 distance vs displacement Displacement is a vector quantity. It is the object's overall change in position. It can be positive or negative depending on your initial point of reference.

38 As any object moves from one position to another, the length of the straight line drawn from its initial position to the object s final position is called displacement. Displacement doesn t always tell you distance an object moved.

39 Displacement is the straight-line separation of two points in a specified direction. D = 12 m, 20 o A q B A vector quantity: contains magnitude AND direction. ex: 12 m, 30 0 ; 8 km/h, N

40 In the diagram below, the origin, or initial position (d o ) is at 0.0 m. The final position (d 1 ) is at 50 m. The distance traveled from the origin is 50 m, but the displacement is 50 m to the right, or can be drawn similar with a green line vector.

41 Defining the reference point and direction If displacement is positive, the object moves to the right. If the displacement is negative, the object moves to the left.

42 If the girl walks to the red ball, then walks backwards to the bear, what distance has she traveled? X Axis Displacement = the distance of a body's change in position from a starting point. Her final displacement is.

43 remember: Displacement (blue line) is how far the object is from its starting point, regardless of how it got there. Distance traveled (dashed line) is measured along the actual path.

44 Physicists use the tools of math to describe measured or Physics shorthand predicted relationships between physical quantities in a situation. difference or change in x, y t sum or total Equation = a compact statement based on a model of the situation. change in position time interval x = x f - x i Shows how 2 or more variables are thought to be related.

45 How are distance and displacement related to motion? Motion of an object is BOTH a scalar quantity (time) And Vector quantity (displacement)

46 If displacement is the distance and direction between 2 positions = change in position = final position initial position x i initial position Displacement: x = x f x i x f final position

47 The values of x i and x f are determined by their positions on the axis. While the choice of a reference point for the coordinate system is arbitrary, once chosen, the same point must be used throughout the problem.

48 Signs of Displacement In physics, the movement from the origin is thought of as positive or negative. In each case, an origin, starting place, or reference point needs to be established. Then, it must be decided which directions are considered positive and which is considered negative. Once a group agrees on that, then you can determine displacement vectors. For example, if we assume that up is positive, then Mt. Elinor, would have an elevation displacement vector of +2,400 ft, based off of the origin of sea level. On the other hand, Death Valley, California is below sea level by 120 ft, so its displacement would be -120 ft compared to sea level.

49 When would position be but the displacement +?

50 displacement is written as: Left: Displacement is positive. Right: Displacement is negative.

51 Check it: In this motion diagram the length of the arrow indicates the change in position of the object, or its. a. Displacement b. Magnitude c. Position d. Resultant

52 In this image, 7 cm is a. a. Vector b. speed c. Scalar d. interval

53 Consider a car that travels 8 m, E then 12 m, W. What is the car s displacement? D = 4 m, W D 8 m,e x What is the distance traveled? 20 m!! x = m,w x = +8

54 try a problem A person starts at the 5.0 m mark. They walk to the 12m mark. What is their distance travelled? What is their displacement? They leave the 12m mark and walk to the 1m mark. What is their distance travelled? What is their displacement? What is the total distance travelled? What is the total displacement for the motion?

55 What about at angles? You drive 3 miles east and then 4 miles north. What is your: Why are you not drawing a picture people? Distance travelled? Displacement?

56 C 4 mi Distance: AB + BC = 7 mi B 3 mi A Displacement: measure of the hypotenuse (AB) 2 + (BC) 2 = (AC) 2 (3) 2 + (4) 2 = x = 25 = 5 mi., NE

57 Try some on your own

58 VELOCITY, SPEED, AND ACCELERATION

59 The Cheetah: A cat that is built for speed. Its strength and agility allow it to sustain a top speed of over 100 km/h. Such speeds can only be maintained for about ten seconds.

60 Speed The average speed of an object is defined as the total distance traveled divided by the total time elapsed Average speed d v t Speed is a scalar quantity total distance total time

61 Speed equation Speed (meters/second) = distance (in meters) time (sec) s = d t Average speed totally ignores any variations in the object s actual motion during the trip. The total distance and the total time are all that is important

62 calculate A s = 20 m B d s = t = 20 m 4 s Time t = 4 s s = 5 m/s Scalar magnitude only, not direction dependent!

63 There are three types of speed: 1) Average speed average of all your speeds over the whole trip. For example, on a trip to Florida, (910 miles) the traveler took 13 hours to get there. The average speed would be 70 mph. This doesn t mean that the car was going exactly 70 mi/hr the entire time. Sometimes the car was going faster, and others slower. 2) Constant Speed - traveling at the same rate for a long period of time. Constant speed is having the cruise control on in the car. The car maintains the same speed the entire time you are clocking it. 3) Instantaneous speed - rate at which an object is traveling at a certain moment. This is your speedometer in your car. It tells you how fast the car is going at the time you look at it.

64 velocity A quantity that measures how fast an object s position has moved from one point to another in a certain direction.

65 Definition of Velocity Velocity is the displacement per unit of time. (A vector quantity.) d = 20 m A D=12 m 20 o B v = 3 m/s at 20 0 N of E Time t = 4 s Direction required!

66 Velocity It takes time for an object to undergo a displacement The average velocity is the rate at which the displacement occurs x xf x V i v= d average = d 1 d 0 t t t t t f 1 t 0 generally use a time interval, so t i = 0 i

67 speed and velocity are NOT the same Velocity the rate at which an object changes its position (has direction) Speed is the magnitude of velocity (how fast an object is moving)

68 The Signs of Velocity Velocity is positive (+) or negative (-) based on direction of motion First choose + direction; then v is positive if motion is with that direction, and negative if it is against that direction.

69 Velocity can be +/- Velocity is a measure of the speed of an object AND the direction it is moving in space. On the escalator, passengers are moving at the same constant speed, but they are moving in different directions. Velocity can change even if speed is remaining constant (you just change direction)

70 Recap: Velocity is defined as a vector quantity that tells the ratio of the displacement change to the time change, or how fast an object is going and in what direction. Speed, on the other hand, is merely the magnitude of the velocity, or how fast an object is moving.

71 Speed vs. Velocity Cars on both paths have the same average velocity since they had the same displacement in the same time interval The car on the blue path will have a greater average speed since the distance it traveled is larger

72 example problem 5.2 m/s =.0052 km x 360 s = 1.87 km During a race, Carla covers 650 m in 125 s running east on a s straight road. hr Find Carla s hr average speed km = 5 km = 2.67 hr s = d = 650 m 5.2 m/s hr x hr t 125 s How long will it take her to run 5 km?

73 Try one Heather and Matthew walk eastward with a speed of.98 m/s. If it takes them 34 min to walk to the store, how far have they walked? Knowns? What do you know? Write it down. Speed =.98 m/s, time = 34 minutes (2040 sec) Unknown? What do you want to know? How far? Distance =? Equation? Write the equation you ll use. Speed = distance / time Work the problem..98 m/s = distance / 2040 sec; d = 2000 meters

74 Example. A runner runs 200 m, east, then changes direction and runs 300 m, west. If the entire trip takes 60 s, what is the average speed and what is the average velocity? Recall that average speed is a function only of total distance and total time: s 2 = 300 m start s 1 = 200 m Total distance: s = 200 m m = 500 m Average speed total path time 500 m 60 s Avg. speed 8.33 m/s Direction does not matter!

75 Example 1 (Cont.) Now we find the average velocity, which is the net displacement divided by time. In this case, the direction matters. v x f t x 0 x f = -100 m t = 60 s x 1 = +200 m x 0 = 0 m; x f = -100 m W x o = 0 E v 100 m 0 60 s 1.67 m/s Direction of final displacement is to the left as shown. Average velocity: v 1.67 m/s, West Note: Average velocity is directed to the west.

76 Example 2. A sky diver jumps and falls for 600 m in 14 s. After chute opens, he falls another 400 m in 150 s. What is average speed for entire fall? Total distance/ total time: 14 s v x t A A x t B B 600 m m 14 s s A 625 m v 1000 m 164 s v 6.10 m/s B Average speed is a function only of total distance traveled and the total time required. 142 s 356 m

77 checking for understanding An Indianapolis 500 car races around the track at 225 mph. At the end of the race (500 miles), what was its average velocity?

78 for example A book gets pushed around the perimeter of a table with dimensions 1.75 m X 2.25 m. It completes this motion in 23 s. What is its average speed? What is its average velocity?

79 another example Car A travels from New York to Miami at a speed of 25 m/s. Car B travels from New York to Canada at a speed of 25 m/s. Are their velocities equal? explain

80 Problems You run down the road 500m. It takes you 32sec to complete the task. What is your: average speed? average velocity? displacement? distance? You run around a circular track (radius of 300m) in 32 sec What is your: average speed? average velocity? displacement? distance? What if you went half way? Will your average speed ever be zero?

81 one more for good measure You travel on a straight highway from your house to visit your friend 370 km (230 mi) to the west. You leave your house at 10 am and arrive at 3 pm. However, after you left your house, you realized you forgot your toothbrush. You were only 15 km down the road so you went back and got it. Half way to your friend s house, you took a short 5 km side road to grab a burger at your favorite burger place. What was your average velocity speed for this trip?

82 Moving Man Graphing Motion

83 acceleration

84 Think about this... What are three ways to change the velocity of a car? Speed up Slow down Change direction

85 Slow your car to a stop at a stop sign. Slow from 9 m/s to 0.0 m/s in 5 s. Slam on the breaks to stop. Slow from 9 m/s to 0.0 m/s in 1.5 s.

86 acceleration The rate at which an object changes its speed or velocity in a given time. If velocity must include a direction (vector). Any time an object s speed or velocity is changing, the object has an acceleration.

87 acceleration = change in velocity = v change in time t v t acceleration = (v f ) - (v2 i ) 1 a time elapsed v t 2 v t 1 If a car is moving at a constant velocity, then its acceleration is zero!

88 calculating average acceleration average acceleration = change in velocity change in time Acceleration is represented in (m/s 2 ) a = v units - v = (m/s) = m X 1 = m t t s s s s 2

89 in one-dimensional motion we only need the sign to show when something is speeding up or slowing down. here is slowing down:

90 - the sign of the velocity and the acceleration is the same if the object is speeding up and - the sign of the velocity and the acceleration is the opposite if the object is slowing down.

91 Review of Symbols and Units Displacement meters (m) Velocity/speed; meters per second (m/s) Acceleration; meters per s 2 (m/s 2 ) Time; seconds (s) Make sure you convert any units when solving problems so they are all the same!

92 try solving Find the acceleration from the first problem. Slow your car to a stop at a stop sign. Slow from 9 m/s to 0.0 m/s in 5 s. a = v - 9 m/s = m /s 2 v f - v i t 5 s t f - t i Slam on the breaks to stop. Slow from 9 m/s to 0.0 m/s in 1.5 s. -9 m/s = m /s s

93 another problem to solve A meteoroid changed velocity from 1.0 km/s to 1.8 km/s in 0.03 seconds. What is its acceleration? (1.8) (1) = 0.03 s 26.7 m/s 2

94 While racing out of our school parking lot, I time myself at a speed of 40 meters per second seven seconds after starting. What was my acceleration during this time? Acceleration = v f v i t 2 t 1 (40 m/s) / 7 s = 5.7 m/s 2 If I were to accelerate at this rate for another ninety seconds, how fast would I be going? a = v/t so v = at (5.7 m/s 2 )(97 sec) = 550 m/s

95 Try some on your own