Unit 2 - Motion. Chapter 3 - Distance and Speed. Unit 2 - Motion 1 / 76

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1 Unit 2 - Motion Chapter 3 - Distance and Speed Unit 2 - Motion 1 / 76

2 Precision and Accuracy Precision is a measure of how closely individual measurements agree with one another. Accuracy refers to how closely individual measurements agree with the correct or true value. Unit 2 - Motion 2 / 76

3 Significant Figures All digits of a measured quantity are called significant figures. Every measurement has uncertainty. Numbers in which there is no uncertainty are called exact numbers and are rare in science. What s the temperature? Unit 2 - Motion 3 / 76

4 Significant Figures All digits of a measured quantity are called significant figures. Every measurement has uncertainty. Numbers in which there is no uncertainty are called exact numbers and are rare in science. What s the temperature? Unit 2 - Motion 3 / 76

5 Determining the Number of Significant Figures: 1 Zeroes between nonzero digits are always significant. Ex: 1005 kg (four sig. figs.); 7.03 cm (three sig. figs.). 2 Zeroes at the beginning of a number are never significant. Ex: 0.02 g (one sig. fig.); cm (two sig. figs.). 3 Zeroes at the end of a number are significant if the number contains a decimal point. If no decimal is present, the trailing zeroes are not significant. Ex: g (three sig. figs.); 3.0 cm (two sig. figs.); (one sig. fig.). Unit 2 - Motion 4 / 76

6 Examples: Determine the number of significant figures in the following measurements cm m m/s km yr s Unit 2 - Motion 5 / 76

7 Dimensional Analysis Dimensional analysis is a systematic way of solving numerical problems that involve the conversion of units. The strategy: Given unit desired unit given unit = desired unit Unit 2 - Motion 6 / 76

8 Examples: Use dimensional analysis to perform the following conversions centimetres to metres metres to kilometres seconds to hours Unit 2 - Motion 7 / 76

9 Practice: Use dimensional analysis to perform the following conversions grams to kilograms litres to millilitres 3 45 minutes to days Unit 2 - Motion 8 / 76

10 Converting Units Using Two or More Conversion Factors 1 The average speed of a nitrogen molecule in air at 25 C is 515 metres per second (m/s). Convert this speed to kilometres per hour. 2 Convert 20 feet per second to miles per hour. [1 mi = 5280 ft] Unit 2 - Motion 9 / 76

11 What is distance? Distance - The amount of space between two objects or points. Measured in: kilometres (km), metres (m), centimetres (cm), or millimetres (mm) Unit 2 - Motion 10 / 76

12 What is time? Time - The duration between two events. Measured in: seconds (s), minutes (min), hours (h), or years (yr) Unit 2 - Motion 11 / 76

13 Position - Is an object s location relative to a reference point. Uniform Motion - Motion at a constant speed in a straight line. Inertia - The resistance of any physical object to any change in its state of motion. Watch: Motion Unit 2 - Motion 12 / 76

14 Investigating Speed, Time, and Distance What is speed? What is the relationship between speed, distance, and time? Problem: Determine the average speed of a motorized toy car. Unit 2 - Motion 13 / 76

15 Relating Speed to Distance and Time Average Speed Speed - the rate (change in time) at which an object covers distance. speed = distance time Average speed - the total distance divided by the total time for a trip. v av = d t v av - average speed d = d 2 d 1 (2nd distance - 1st distance) t = t 2 t 1 (end time - start time) Unit 2 - Motion 14 / 76

16 Solve for the missing variable: 1 d = 20.5 km; t = 5.5 h 2 t = 15.5 h; v av = 65.0 km/h 3 d = 155 km; v av = 110 km/h Unit 2 - Motion 15 / 76

17 Example 1. Jennifer walks to school a total distance of 4.5 km. The trip takes her 45.5 minutes. What was her average speed in km/h? Unit 2 - Motion 16 / 76

18 Example 2. You are on a train and you see a sign that reads 120 km. You decide to measure the amount of time it takes to go from one sign to another. If the signs are 10 km apart and it takes 391 s to travel between them, how fast is the train going in km/h? Unit 2 - Motion 17 / 76

19 Example 3. Kara can predict how long it will take to bike from her house to the beach. The distance is 45 km and she can bike at 20 km/h. How long will the trip take her? Unit 2 - Motion 18 / 76

20 Example 4. As a summer job, Mike analyzes grazing patterns of a herd of bison. He notes that they graze at about 110 m/h for about 7.0h/d. What distance (in km) will the herd have travelled in two weeks (14 d). Unit 2 - Motion 19 / 76

21 Distance - Time Graphs Graphs help us understand the relationship between two variables - an independent and a dependent variable. Unit 2 - Motion 20 / 76

22 In Distance - Time graphs: Independent variable - time Dependent variable - distance We can use these graphs to determine speed. Which line below represents the fastest speed? Unit 2 - Motion 21 / 76

23 Recall: y = mx + b is the general equation for a straight line. y - dependent variable x - independent variable m - slope of the line b - y intercept of the line Question: How can the slope of the best-fit line represent both v av = d t and y = mx + b? Unit 2 - Motion 22 / 76

24 Example: The motion of two bicycle riders, Tom and Jerry, is described on a distance - time graph. 1 From just looking at the graph, which rider has the greater speed? 2 Calculate the speed of each rider by determining the slope of each line Unit 2 - Motion 23 / 76

25 Practice: Sketch a distance - time graph for a car that starts at rest and reaches a final speed of 80 km/h. Unit 2 - Motion 24 / 76

26 Example: A car is travelling across the Confederation Bridge. The distances and times are listed below. They include a short stretch of road beyond the end of the 12.9 km bridge. 1 Plot a distance - time graph. Draw a best-fit straight line. 2 Using the graph, find the distance travelled after 5.0 min. 3 What is the required time to cross the bridge? 4 What is the speed of the car in kilometres per hour? Time (min) Distance (km) Unit 2 - Motion 25 / 76

27 Unit 2 - Motion 26 / 76

28 Chapter 4 - Displacement and Velocity What is acceleration? What is the relationship between acceleration, speed, and time? Problem: Determine at what time the tennis ball is accelerating, decelerating, or neither. Determine the formula for calculating acceleration. Unit 2 - Motion 27 / 76

29 When speed is not constant, it may change slowly or rapidly. Acceleration (a) is the rate of change in speed. It is calculated using the following formula: a av = v t If the change in speed is the same in equal intervals of time, then this is called constant acceleration. Average acceleration (a av ) is the average rate of change in speed of an object. Watch: Car Accelerating Watch: X2 (on-ride) Watch: X2 (off-ride) Watch: KingDa Ka (on-ride) Unit 2 - Motion 28 / 76

30 What are the units for acceleration? Example: v = km/h t = h = a av = km/h h = km/h 2 Unit 2 - Motion 29 / 76

31 What are the units for acceleration? Example: v = km/h t = h = a av = km/h h Examples: = km/h 2 1 v = 105 km/h t = h a av =? 2 v =? t = 1.5 h a av = 25 km/h 2 3 v = 13 m/s t =? a av = 8 m/s 2 Unit 2 - Motion 29 / 76

32 Example 1: A powerful car can accelerate from 0 to 100 km/h in 6.0 s. What is its average acceleration? [No conversions are required]. Unit 2 - Motion 30 / 76

33 Example 2: Myriam Bedard (Olympic skier) accelerates at an average 2.5 m/s 2 for 1.5 s. What is her average speed (in m/s) at the end of 1.5 s? Unit 2 - Motion 31 / 76

34 Example 3: A skateboarder rolls down a hill and changes his speed from rest to 1.9 m/s. If the average acceleration down the hill is 0.40 m/s 2, for how long (in s) was the skateboarder on the hill? Unit 2 - Motion 32 / 76

35 Refining the Acceleration Equation Often, when acceleration is involved, the initial speed is known. The initial speed is often a nonzero value; hence, speed can be more formerly defined as: v = v 2 v 1 Then, our acceleration formula is: a av = v 2 v 1 t Unit 2 - Motion 33 / 76

36 Example 4: Kerrin is moving at 1.8 m/s near the top of a hill. 4.2 s later she is travelling at 8.3 m/s. What is her average acceleration? Unit 2 - Motion 34 / 76

37 Example 5: A bus with an initial speed of 12 m/s accelerates at 0.62 m/s 2 for 15 s. What is the final speed of the bus? Unit 2 - Motion 35 / 76

38 Example 6: A snowmobile reaches a final speed of 22.5 m/s after accelerating at 1.2 m/s 2 for 17 s. What was the initial speed of the snowmobile? Unit 2 - Motion 36 / 76

39 Example 7: In a race, a car travelling at 100 km/h comes to a stop in 5.0 s. What is the average acceleration? Unit 2 - Motion 37 / 76

40 Speed - Time Graphs for Acceleration Recall: Acceleration is basically a change in speed over time. In speed-time graphs, the following properties are important: slope = rise run = speed time area under the line = distance travelled during that time interval Unit 2 - Motion 38 / 76

41 Practice: Find the area under the following curves. Unit 2 - Motion 39 / 76

42 Example 1: A boat on the St. Lawrence River travels at full throttle for 1.5 h. From the area under the line of the speed-time graph, determine the distance travelled. What was the average acceleration of the boat? Unit 2 - Motion 40 / 76

43 Example 2: Galileo rolls a ball down a long grooved inclined plane. According to a speed-time graph, what is the distance travelled in 6.0 s? What was the average acceleration? Unit 2 - Motion 41 / 76

44 Example 3: This is the speed-time graph of a train s journey. What is the total distance travelled by the train? Unit 2 - Motion 42 / 76

45 Instantaneous Speed Instantaneous Speed is the speed at a particular moment in time. Example: The reading on a speedometer. Note: For any object moving at a constant speed, the instantaneous speed is the same at any time, and equals the constant speed. Unit 2 - Motion 43 / 76

46 Comparing Graphs: Constant speed on Distance -Time graphs Constant speed on Speed -Time graphs Unit 2 - Motion 44 / 76

47 Now suppose that speed is not constant; i.e. acceleration or deceleration occurs. How do we find instantaneous speed? We sketch the tangent to a point on a line. A tangent is a straight line that just touches a curve at one point. This allows us to then calculate the slope of this line at this one point. Unit 2 - Motion 45 / 76

48 Example: Given the Distance (m) vs Time (min) graph below, answer the following questions. Unit 2 - Motion 46 / 76

49 1) Which graph illustrates a constant speed for the whole trip? 2) Which graph shows a constantly changing speed? 3) Which graph(s) have an instantaneous speed of zero at some point? 4) What is the instantaneous speed at 3.5 min for each graph? 5) Calculate the average speed for 0 to 5.0 min for each cyclist. Unit 2 - Motion 47 / 76

50 Practice: What is the instantaneous speed at 2 seconds (time is on the bottom)? What is the instantaneous speed at the point (1, 3)? Unit 2 - Motion 48 / 76

51 Chapter 5 - Displacement and Velocity Vectors: Position and Displacement Reporting distances and speed without a direction is often not very useful. All distances and directions are generally stated relative to a reference point (origin or starting point). Your position is the separation and direction from a reference point. Figure: When we travel, Regina is our reference point. Unit 2 - Motion 49 / 76

52 We will describe distances and direction together using vectors. A vector quantity is a quantity that involves a direction, such as position. Examples: 2 km [E], 73 m [N], or 292 km [S]. Representation: Unit 2 - Motion 50 / 76

53 Displacement Distance is a scalar quantity (involves only size). Displacement is defined as the change in position. Symbol: d Displacement is usually calculated by the following: d = d 2 d 1 Unit 2 - Motion 51 / 76

54 Example: The Roughriders move on a straight downfield pass from the Argonauts 45-yard line to the Argonauts 20-yard line. Then (in classic fashion), lose 5 yards on the next running play. Using the Argonauts 45-yard line as the reference point, what is the ball s: a) position after the pass? c) final displacement b) final position? d) total distance travelled? Unit 2 - Motion 52 / 76

55 Adding Vectors Along a Straight Line We can add straight-line vectors by either drawing a vector diagram or by using arithmetic. We must draw our vectors to scale. Note: d R - denotes the resultant vector Example 1: Anne takes her dog for a walk. They walk 250 m [W] and then back but only 215 m [E]. Find their resultant displacement. Unit 2 - Motion 53 / 76

56 Note: When computing vector addition algebraically, let [E] be positive and [W] be negative. Then, d R = d 1 + d 2 Example 2: Anne goes for another walk. She leaves home and walks 250 m [W] and then back back 175 m [E]. Find d R. Unit 2 - Motion 54 / 76

57 Example 3: Anne walks 30 m [W], stops to chat, then continues 50 m [W] before returning 60 m [E]. Find d R. Unit 2 - Motion 55 / 76

58 Adding Vectors at an Angle Displacement can occur in all directions. We will use angles to specify most displacements. Method: If the direction that we re interested in does not exactly match one of the compass points, we write it as an angle from the closest compass point. Examples: a) [30 E of S] b) [10 E of N] Unit 2 - Motion 56 / 76

59 Example 1: Denise walks to Sarah s home by going one block east and then one block north. Each block is 160 m long. What is Denise s final displacement? Unit 2 - Motion 57 / 76

60 Example 2: This time, Denise walks 180 m east to get to Sarah s home. Confused, she then walks 150 m west. What is Denise s final displacement? Unit 2 - Motion 58 / 76

61 Velocity Speed is a very common quantity (car speed, airplane speed, etc.). But often speed is only useful when associated with some direction (wind speed direction, airplane flight patterns, etc.). Velocity is a speed along with a direction. v = d t or v av = d R t where, d or d R - change in displacement or resultant displacement t - change in time v or v av - velocity or average velocity Unit 2 - Motion 59 / 76

62 Example 1: A train travels at a constant speed through the countryside and has a displacement of 150 km [E] in a time of 1.7 h. What is the velocity of the train? Unit 2 - Motion 60 / 76

63 Example 2: Monarch butterflies migrate from Eastern Canada to central Mexico, a resultant displacement of about 3500 km [SW] in a time of about 91 d. What is their average velocity in km/h? Unit 2 - Motion 61 / 76

64 Practice: A monarch butterfly usually flies during the day and rests at night. If its velocity is 19 km/h [S] for 230 km [S] on one part of its journey to Mexico, how long does this take? Unit 2 - Motion 62 / 76

65 Example 3: A student travels 6.0 m [E] in 3.0 s and then 10.0 m [N] in 4.0 s. Calculate the student s average velocity. (Hint: Draw a vector diagram to determine the resultant displacement). Unit 2 - Motion 63 / 76

66 Practice 1: A car travels 8.0 km [N] and then turns suddenly west and travels an additional 6.0 km. When the car stops, it has travelled a total time of 20.5 minutes. Calculate the car s average velocity. Unit 2 - Motion 64 / 76

67 Practice 2: A cougar moves rapidly 2.0 km [S] in 5.0 minutes and then makes a sudden turn to the east 4.0 km, which takes 12.0 minutes. Determine the cougar s average velocity. Unit 2 - Motion 65 / 76

68 Chapter 6 - Velocity and Acceleration Position - Time Graphs What s the difference between a position - time graph and a distance - time graph? Shows motion with constant speed Shows motion eastward with constant velocity Unit 2 - Motion 66 / 76

69 Example 1: When Donovan Bailey finished the 4 x 100 m relay race, the team s time was s, as describe graphically in the figure below. Determine Bailey s velocity algebraically and graphically. Unit 2 - Motion 67 / 76

70 Note: The slope of the tangent at a point on a position - time graph yields the instantaneous velocity. Example 2: A boat accelerates uniformly for seven seconds. What is the instantaneous velocity at 4.9 s? Assume that east is positive. Unit 2 - Motion 68 / 76

71 Example 1: From the graph below, describe what is occurring in each segment. Unit 2 - Motion 69 / 76

72 Velocity - Time Graphs Recall: a = v t = on a velocity - time graph, slope = acceleration. Example 2: What is the acceleration of the diving kingfisher (shown in the graph below)? Note: Up is the positive direction and down is negative. Unit 2 - Motion 70 / 76

73 Acceleration and Velocity Recall from Chapter 2: a = v 2 v 1 t Vector acceleration is simply change in velocity in a given time: a = v 2 v 1 t Unit 2 - Motion 71 / 76

74 Example 1: Suppose a plane (taking off) starts from rest and accelerates to a final velocity of 270 km/h [E] in a time of 32 s. Calculate the acceleration of the airplane. Assume east is positive. Watch: Very fast take off Unit 2 - Motion 72 / 76

75 Example 2: Suppose the same plane reaches its destination and touches down on the runway travelling at 305 km/h [E]. If the plane takes 25 s to come to a complete stop, what is its vector acceleration? Unit 2 - Motion 73 / 76

76 Example 3: An air puck on an air table is attached to a spring. The puck is fired across the table at an initial velocity of 0.45 m/s [right] and the spring accelerates the air puck at an average acceleration of 1.0 m/s 2 [left]. What is the velocity of the air puck after 0.60 s? Assume that right is positive and left is negative. Unit 2 - Motion 74 / 76

77 Investigating Acceleration due to Gravity What is the acceleration of falling objects? Problem: The acceleration of gravity was first discovered by Isaac Newton. We will attempt to make him proud by discovering what this acceleration is. We will use his formula to help us: a g = 2h t 2 a g - acceleration due to gravity h - height t - total time where, Unit 2 - Motion 75 / 76

78 Example 1: A person throws a ball straight up from the ground. The ball leaves the person s hand with an initial velocity of 10.0 m/s [up]. Assume up is positive and down is negative. a) What is the velocity of the ball after 0.50 s? b) What is the velocity of the ball after 1.50 s? Unit 2 - Motion 76 / 76