Graphing Motion. Every Picture Tells A Story

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1 Graphing Motion Every Picture Tells A Story

2 Read and interpret motion graphs Construct and draw motion graphs Determine speed, velocity and accleration from motion graphs

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4 If you make a graph by hand it should always be on graph paper. The graph should fill the available space. Carefully choosing the best scale is necessary to achieve this.

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6 The graph should always have a title.

7 Always label the x and y axes in 3 ways: title, numerical values, and units.

8 Always make a line graph line graphs are way more handy, because they tell you how one thing changes under the influence of some other variable.

9 The x axis is always the independent variable. If time is one of the measurements being graphed, it always goes on the x-axis. Independent Variable or Manipulated Variable is what you are testing. It is what causes things to change as you make changes to it. Some people nickname it the I-do variable.

10 The y axis is always the dependent variable. Y axis Dependent Variable or the Responding Variable is the effect and it may or may not change. It is observed during as well as at the end of the experiment.

11 Dependent Responing Y-axis Manipulated Independent X-axis D = dependent variable R = responding variable Y = graph information on the vertical axis M = manipulated variable I = independent variable X = graph information on the horizontal axis

12 Example of a Bad Graph There's no title. What's it a graph of? Who knows? There are no labels on the x or y axis. What are those numbers? Who knows? There are no units on the x or y axis. Is this a graph of speed in miles per hour or a graph of temperature in Kelvins? Who can tell?

13 What s wrong with this graph? There's no title. What's it a graph of? Who knows? There are no increments on the axes, and there are no gridlines. There are no labels on the x or y axis. What are those numbers? Who knows? There are no units on the x or y axis. What size are the numbers: kilo-, centi-, or milli-?

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15 Definition of slope Numerical measure of a line's incline relative to the horizontal. In analytic geometry, the slope of any line, ray, or line segment is the ratio of the vertical to the horizontal distance between any two points on it ( slope equals rise over run ).

16 A displacement time graph with time along x axis and displacement on the y axis. Velocity is positive when the object moves in the positive direction. Since the position value is increasing, the graph slopes upward. Zero slope Velocity is zero when the position does not change. This line has zero slope. Velocity is negative when the object moves In the negative direction on the Axis system. This line has a negative Slope.

17 Types of Motion Graphs Distance vs Time Position vs Time Velocity vs Time Acceleration vs Time

18 Distance vs Time Graphs Speed is the distance an object travels per unit of time. You can graphically represent the speed or an object using a distance-time graph.

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21 If the speed is constant, then the slope is constant (straight line).

22 Constant Speed A uniform distance is covered for each unit of time. A constant speed graph shows a constant & positive slope

23 The steeper the slope, the faster the speed.

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25 If the speed is changing, then the slope is changing (curve).

26 Describe the motion in each section of the graph. Decelerating Stopped Accelerating Steady speed

27 Position vs time Graphs of Constant motion Position vs. time graphs give you an easy and obvious way of determining an object s displacement at any given time, and a subtler way of determining that object s velocity at any given time.

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29 A position-time graph, is one in which position is Plotted on the y-axis and the time is on the x-axis. A position-time graph is similar to a distance-time Graph but has direction on the y-axis.

30 Although distance-time and position-time graphs can show Similar graphs, this is not always the case. Below is a graph of a person who walked to a nearby store (10 km north) and back to the original reference point, this would mean the total travelled distance is 20 km (10 km to the store and 10 km back). The distance-time graph is on the left.

31 The position-time graph looks different because the position changed when the person turned back from the store back to the original reference point.

32 Looking at the slope of a distance vs time or a position vs time graph... Slope = Velocity As slope goes, so does velocity. If the speed is constant, then the slope is constant (straight line). If the speed is changing, then the slope is changing (curve). If the velocity is positive, then the slope is positive (moving upward, towards the right). If the velocity is negative, then the slope is negative (moving downward, towards the right). The steeper the line/curve, the faster the speed.

33 Reading and interpreting position-time graphs

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35 s vs t - The object is standing still at a positive location. Time is going by but the position is not changing. Since the slope equals zero it has no movement.

36 s vs t - the object is traveling at a constant positive velocity. The locations of its position are increasingly positive.

37 s vs t - the object is traveling at a constant positive velocity but is traveling through a negative region.

38 s vs t - this slope represents a constant negative velocity since the object is traveling in a negative direction at a constant rate. Notice that the locations of its position are becoming less and less positive

39 s vs t - the object is traveling at a constant negative velocity through a negative region. The locations of its position are increasingly negative.

40 The meaning of slope on a position-time graph! If calculated properly, it shows the velocity of the motion.

41 In this graph Car A moves for 5 seconds a distance of 10 meters. How can we figure out the velocity of the car from the graph? We can use the formula for the slope of a line to get the velocity.

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43 Any point on this graph shows the position of the ant at a particular moment in time.

44 The point at (2, 2) show that, two seconds after it started moving, the ant was two centimeters to the left of its starting position. The point at (3,1) shows that, three seconds after it started moving, the ant is one centimeter to the right of its starting position.

45 For the first two seconds, the ant is moving to the left. The next second, it reverses its direction and moves quickly to y = 1. The ant then stays still for three seconds before it turns left again and moves back to where it started.

46 For any position vs. time graph, the velocity at time t is equal to the slope of the line at time t. In a graph made up of straight lines, like the one for the ant, the slope can easily be calculated at each point on the graph to show the instantaneous velocity at any given time.

47 Determine the ant s instantaneous velocity at any given point during the trip. Remember the instantaneous velocity shows the velocity of the ant at one point. The ant is cruising along at the fastest speed between t = 2 and t = 3, because the position vs. time graph is steepest between these points.

48 Calculate the ant s average velocity during this time interval is a simple matter of dividing rise by run. Remember average velocity is the total displacement divided by the total time. The average velocity here is zero because the total diaplacement is zero. 0/7 = 0 m/s

49 Stage 1: The car moves forwards from the origin to in the first 5 s. Calculate the velocity for the car after the first five seconds.

50 Stage 2: The car moves backwards, passes the origin, to in the next 5 s. Calculate the velocity of the car between five and ten seconds.

51 Stage 3: The car remains at rest in the last 5 s. Calculate the velocity of the car for the last five seconds.

52 Distance (km) Different Slopes Slope = Rise/Run = 1 km/1 hr = 1 km/hr Run = 1 hr Slope = Rise/Run = 0 km/1 hr = 0 km/hr Rise = 1 km Run = 1 hr Rise = 0 km Time (hr) Run = 1 hr Rise = 2 km Slope = Rise/Run = 2 km/1 hr = 2 km/hr

53 Position Time Graphs of Accelerated motion Position vs. time graphs give you an easy and obvious way of determining an object s displacement at any given time, and a subtler way of determining that object s velocity at any given time.

54 A very useful aspect of these graphs is that the area under the v-t graph tells us the distance travelled during the motion.

55 Since the slope represents the speed, if the speed is increasing over time, the slope must be also be increasing over time. The graph is a curve that gets steeper as you move along The x-axis. A position-time graph for a ball in free fall is shown below.

56 The graph of an object slowing down is also cuved. The example below show the position-time graph for a car coming to a gradual stop at a red l ight. As time passes, the car s speed decreases. The slope must therefore decrease.

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58 answers

59 Velocity vs Time Graphs d slope = velocity t slope = acceleration v area = distance t a area = velocity t

60 If the graph is a horizontal line, there is no change in velocity, therefore there is no acceleration (the slope is 0). If the acceleration is positive then the slope is positive (the line moves upward to the right). If the acceleration is negative, then the slope is negative (the line moves downward to the right).).

61 Calculating acceleration from a velocity-time graph

62 Calculating the distance on velocity-time graph.

63 An object is moving in the positive direction if the line is located in the positive region of the graph (whether it is sloping up or sloping down). An object is moving in the negative direction if the line is located in the negative region of the graph (whether it is sloping up or sloping down). If a line crosses over the x-axis from the positive region to the negative region of the graph (or vice versa), then the object has changed directions.

64 The object moves in the + direction at a constant speed - zero acceleration (interval A). The object then continues in the + direction while slowing down with a negative acceleration (interval B). Finally, the object moves at a constant speed in the + direction, slower than before (interval C).

65 The object moves in the + direction while slowing down; this involves a negative acceleration (interval A). It then remains at rest (interval B). The object then moves in the - direction while speeding up; this also involves a negative acceleration (interval C).

66 The object moves in the + direction with a constant velocity and zero acceleration (interval A). The object then slows down while moving in the + direction (i.e., it has a negative acceleration) until it finally reaches a 0 velocity (stops) (interval B). Then the object moves in the - direction while speeding up; this corresponds to a - acceleration (interval C).

67 a plot of velocity versus time can also be used to determine the displacement of an object. The diagram below shows three different velocity-time graphs; the shaded regions between the line and the timeaxis represents the displacement during the stated time interval.

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71 The velocity-time graph for a two-stage rocket is shown below. Use the graph and your understanding of slope calculations to determine the acceleration of the rocket during the listed time intervals. When finished, click the buttons to see the answers. 40 m/s 2 20 m/s 2-20 m/s 2

72 Constant positive (rightward) velocity

73 Constant negative (leftward) velocity

74 Rightward velocity with rightward acceleration.

75 Rightward Velocity and negative acceleration

76 Leftward velocity, leftward acceleration

77 Leftward velocity rightward acceleration

78 Acceleration Acceleration the rate at which velocity is changing Acceleration = v/ t Can increase or decrease (sometimes called deceleration) Think of traveling in a car, you can feel the acceleration 3 ways to accelerate in a car 1. Brake pedal slowing down; coming to a stop (changing speed) 2. Steering wheel going around a corner or curve (changing direction) 3. Gas pedal leaving from a stopped position (changing speed)

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80 The object moves in the + direction at a constant speed - zero acceleration (interval A). The object then continues in the + direction while slowing down with a negative acceleration (interval B). Finally, the object moves at a constant speed in the + direction, slower than before (interval C).

81 The object moves in the + direction at a constant speed - zero acceleration (interval A). The object then continues in the + direction while slowing down with a negative acceleration (interval B). Finally, the object moves at a constant speed in the + direction, slower than before (interval C).

82 The object moves in the + direction while slowing down; this involves a negative acceleration (interval A). It then remains at rest (interval B). The object then moves in the - direction while speeding up; this also involves a negative acceleration (interval C).

83 Zero to 90s - On this graph we see a horizontal line that reads 5m/s for those same first 90 seconds. On a v-t graph a flat line means constant velocity. Constant velocity means zero acceleration.

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85 Graphs of Motion Uniform Velocity The area under a velocity vs time graph is the displacement of the object. Find the distance traveled by each object.

86 Acceleration Suppose you are traveling in a car and your speed goes from 10.km/h to 60.km/h in 2.0s. What is your acceleration? Suppose a car goes from 80.km/h to 15km/h in 5.0 seconds. What is the acceleration? A car is coasting backwards down a hill at a speed of 3.0m/s when the driver gets the engine started. After 2.5s, the car is moving uphill at 4.5m/s. Assuming that uphill is in the positive direction, what is the car s average acceleration?

87 Graphs of Motion Velocity vs time graphs: How can you tell if the object is accelerating or decelerating? Accelerating (speeding up) when the magnitude of the velocity is increasing Decelerating (slowing down) when the magnitude of the velocity is decreasing

88 Graph Practice

89 Which pair of graphs shows the same motion? Answer 1

90 pc.org/~clement/simulations/physlets/tst/position- Time%20Graphs.html

91 anics/kin/motion_graph/x-t02_e.html

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93 Stage 1: The car moves forwards from the origin to in the first 5 s.

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95 Stage 2: The car moves backwards, passes the origin, to in the next 5 s.

96 Stage 3: The car remains at rest in the last 5 s.

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106 What is the velocity for each stage of the journey? b. What is the average (mean) velocity for the whole journey

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112 Distance or Displacement Distance how far an object has traveled Indianapolis is about 45 miles away The distance to Indianapolis is 45 miles; the distance back to Bloomington is 45 miles the total distance traveled round trip is 90 miles Displacement how far an object is from its original position (direction matters) The displacement to Indianapolis is 45 miles north; the displacement back to Bloomington is 45 miles south the total displacement is 0 miles You can find displacement by Finding the area under a velocity time graph Using the equation d = v avg * t

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116 Understanding the Connection Between Slope and Velocity The slope of a line for a distance vs. time graph represents the velocity for the object in motion. Slope can be determined using the following formula: The change in y values divided by the change in x values determines the average velocity for the object between any two points.

117 Pick two points on the line and determine their coordinates. Determine the difference in y-coordinates of these two points (rise). Determine the difference in x-coordinates for these two points (run). Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope).

118 rise over run Calculate the velocity between 3 and 4 seconds. Note: This is a constant speed graph, so the velocity should be the same at all points.

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M1. (a) (i) 4.5 allow 1 mark for correct substitution i.e. 9 2 2

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