P.1 Book 2 Section 2.1 Graphs of straight-line motion 2.1Graphs of straight-line motion The Hare and the Tortoise Motion graphs Check-point 1 Check point 2 Graphs for uniformly accelerated motion Other motion graphs Check-point 3 Motion analyzing tools Check-point 4

P.2 Book 2 Section 2.1 Graphs of straight-line motion The Hare and the Tortoise The details of the race are as follows: Time / s Displacement / m TortoiseHare Example 1

P.3 Book 2 Section 2.1 Graphs of straight-line motion The Hare and the Tortoise What does the data tell us about their performances? The tortoise has a larger displacement than the hare.

P.4 Book 2 Section 2.1 Graphs of straight-line motion 1 Motion graphs Common motion graphs: 1Displacement–time graph 2Velocity–time graph 3Acceleration–time graph

P.5 Book 2 Section 2.1 Graphs of straight-line motion 1 Motion graphs Consider the following uniform motion: Take the direction to the right as +ve. Velocity of car = 10 m s –1 Displacement of the car at each second

P.6 Book 2 Section 2.1 Graphs of straight-line motion 1 Motion graphs a Displacement-time graph A displacement–time graph, or s-t graph, shows the displacement of an object at different instants. x-axis: time y-axis: displacement Simulation 2.1 Displacement–time graphs

P.7 Book 2 Section 2.1 Graphs of straight-line motion a Displacement-time graph In the previous example, the s-t graph of the car is a straight line. +ve constant velocity straight line with +ve slope

P.8 Book 2 Section 2.1 Graphs of straight-line motion a Displacement-time graph Slope of displacement-time graph The slope of an s-t graph gives the velocity. Slope of s-t graph = change in displacement change in time = velocity Unit of the slope = unit of displacement unit of time = m s = m s –1

P.9 Book 2 Section 2.1 Graphs of straight-line motion a Displacement-time graph In the example, Slope = change in s change in t = 10 m s –1 = velocity of the car Sign of slope: direction of velocity Magnitude of slope: magnitude of velocity Example 1 Finding velocity from the s-t graph

P.10 Book 2 Section 2.1 Graphs of straight-line motion Example 1 Finding velocity from the s-t graph Refer to. Take the direction towards the goal as +ve. (a)Plot the s-t graph of the hare.

P.11 Book 2 Section 2.1 Graphs of straight-line motion Example 1 Finding velocity from the s-t graph (b)Find the velocity of the hare at t = 170 s from the s-t graph. Velocity = slope of s-t graph at t = 170 s = 50 – – 150 = –0.14 m s –1

P.12 Book 2 Section 2.1 Graphs of straight-line motion a Displacement-time graph Example 2 Analyzing the s-t graph

P.13 Book 2 Section 2.1 Graphs of straight-line motion Example 2 Analyzing the s-t graph Describe the motion of the hare during 0–150 s, 150–200 s and 200–300 s. Time / sv (slope of graph) / m s –1 Motion 0– – –300 v = = 0.38 v = 50 – – 150 = –0.14 v = – 200 = m s –1 towards the goal 0.14 m s –1 towards the starting point at rest

P.14 Book 2 Section 2.1 Graphs of straight-line motion Check-point 1 A boy is walking along a straight road. Here is his s-t graph.

P.15 Book 2 Section 2.1 Graphs of straight-line motion Check-point 1 Complete the following table. Time / s Displacement / m v / m s –1 0– – – – –1

P.16 Book 2 Section 2.1 Graphs of straight-line motion b Velocity-time graph A velocity-time graph, v-t graph, gives the velocity of an object at different times. e.g. a car moving at 10 m s –1 constantly horizontal straight line Simulation 2.2 Velocity-time graphs

P.17 Book 2 Section 2.1 Graphs of straight-line motion b Velocity-time graph i Slope of velocity-time graph The slope of a v-t graph gives the acceleration. Slope of v-t graph = change in velocity change in time = acceleration

P.18 Book 2 Section 2.1 Graphs of straight-line motion i Slope of velocity-time graph In the last example, Sign: direction of acceleration Magnitude: magnitude of acceleration slope of v-t graph = 0 acceleration = 0 (constant velocity) Example 3 Finding acceleration from the v-t graph

P.19 Book 2 Section 2.1 Graphs of straight-line motion Example 3 Finding acceleration from the v-t graph Amy rides on a bus along a straight road. What are her accelerations during 0–10 s, 10–15 s and 15–30 s?

P.20 Book 2 Section 2.1 Graphs of straight-line motion Example 3 Finding acceleration from the v-t graph Acceleration a = slope of graph 0–10 s: a = 0 10–15 s: a = 17 – – 10 = 1 m s –2 a = 14 – – 15 15–30 s: = –0.2 m s –2 (uniform velocity) (accelerate) (decelerate)

P.21 Book 2 Section 2.1 Graphs of straight-line motion b Velocity-time graph ii Area under velocity-time graph The area under a v-t graph is the total displacement during the time interval. Area above the time axis: +ve Area below the time axis: –ve Sign of area: direction of displacement

P.22 Book 2 Section 2.1 Graphs of straight-line motion v t tt v1v1 v2v2 v3v3 v4v4 t t tt tt ii Area under velocity-time graph A 1 = v 1  t = s 1 A 2 = v 2  t = s 2 A 3 = v 3  t = s 3 A 4 = v 4  t = s 4

P.23 Book 2 Section 2.1 Graphs of straight-line motion v ii Area under velocity-time graph A1A1 A2A2 A3A3 A4A4 t Area under each strip =displacement made under a uniform velocity in  t Total displacement = s 1 + s 2 + s 3 + s 4 = A 1 = area under graph + A 2 + A 3 + A 4 Example 4 Finding displacement from the v-t graph

P.24 Book 2 Section 2.1 Graphs of straight-line motion Example 4 Finding displacement from the v-t graph Consider the coloured area under the graph. Find the total displacement during 0–3 s. Total displacement = area under v-t graph = 10  3 = 30 m

P.25 Book 2 Section 2.1 Graphs of straight-line motion b Velocity-time graph Example 5 Drawing a v-t graph

P.26 Book 2 Section 2.1 Graphs of straight-line motion Example 5 Drawing a v-t graph John goes from O to B at 1 m s –1 for 200 s, stays there for 100 s and returns to A at 2 m s –1 for 500 s.

P.27 Book 2 Section 2.1 Graphs of straight-line motion Example 5 Drawing a v-t graph Take the direction to the right as +ve. (a)Draw the v-t graph of the whole journey.

P.28 Book 2 Section 2.1 Graphs of straight-line motion Example 5 Drawing a v-t graph (b)Find the displacement from the graph in (a) during (i)0–200 s; (ii)200–300 s; (iii)300–400 s; (iv)400–800 s.

P.29 Book 2 Section 2.1 Graphs of straight-line motion Example 5 Drawing a v-t graph Time / s 0– – – –800 Displacement s = area under graph = v  change in time s = 1  (200 – 0) = 200 m s = 0  (300 – 200) = 0 s = (–2)  (400 – 300) = –200 m s = (–2)  (800 – 400) = –800 m moves from O to B stays at B moves from B to O moves from O to A

P.30 Book 2 Section 2.1 Graphs of straight-line motion Check-point 2 – Q1 Here is the motion of a car: 1Travels at constant velocity of 10 m s –1 for 15 s 2Then slows down at 2 m s –2 to a stop Sketch the v-t graph of the car. Take the forward direction as +ve.

P.31 Book 2 Section 2.1 Graphs of straight-line motion Check-point 2 – Q1 t / s v / m s  Take the forward direction as +ve. uniform velocity deceleration

P.32 Book 2 Section 2.1 Graphs of straight-line motion Check-point 2 – Q2 A car moving on a road has an oil leak. Its v-t graph is shown below. Oil drips out drop by drop steadily and leaves some dirt marks on the road.

P.33 Book 2 Section 2.1 Graphs of straight-line motion C Check-point 2 – Q2 Which of the dirt mark patterns is correct? A B

P.34 Book 2 Section 2.1 Graphs of straight-line motion Check-point 2 – Q3 In a race, two runners X and Y start at the same place at time t = 0. Their v-t graphs are shown below.

P.35 Book 2 Section 2.1 Graphs of straight-line motion Check-point 2 – Q3 Which of the following pictures correctly shows their positions at time t = 10 s? A B C

P.36 Book 2 Section 2.1 Graphs of straight-line motion c Acceleration-time graph An acceleration-time graph, a-t graph, gives the acceleration of an object at different instants. A car moving at 10 m s –1 (constant velocity): acceleration = 0

P.37 Book 2 Section 2.1 Graphs of straight-line motion a / m s –1 t / s c Acceleration-time graph The s-t, v-t and a-t graphs are interrelated: Velocity = slope of s-t graph Displacement = area under v-t graph Acceleration = slope of v-t graph v / m s –1 t / s s / m s –1 t / s

P.38 Book 2 Section 2.1 Graphs of straight-line motion 2 Graphs of uniformly accelerated motion a Uniformly accelerated motion in one direction Consider the following motion: Take the direction to the right as +ve. Acceleration = 2 m s –2

P.39 Book 2 Section 2.1 Graphs of straight-line motion a Uniformly accelerated motion in one direction Velocity increases by 2 m s –1 per second The motion graphs of the car: Area under the graph = displacement Slope of the graph = acceleration

P.40 Book 2 Section 2.1 Graphs of straight-line motion a Uniformly accelerated motion in one direction Relation between the s-t, v-t and a-t graphs: s-t graph v-t graph a-t graph slope area Simulation 2.3 Relation between motion graphs

P.41 Book 2 Section 2.1 Graphs of straight-line motion b Uniformly accelerated motion with a change in direction If the acceleration and the initial velocity are in opposite directions, the object will: slow down momentarily at rest speed up in the direction of acceleration

P.42 Book 2 Section 2.1 Graphs of straight-line motion b Uniformly accelerated motion with a change in direction Take the direction to the right as +ve. A car moves towards the right at 10 m s –1 and accelerates towards the left at 2 m s –2. s-t graph v-t graph a-t graph

P.43 Book 2 Section 2.1 Graphs of straight-line motion b Uniformly accelerated motion with a change in direction Simulation 2.4 Motion graphs 2.5 Draw your own v-t graphs Simulation

P.44 Book 2 Section 2.1 Graphs of straight-line motion 1 m s –1 for 200 s 3 Other motion graphs Distance-time and speed-time graphs can also be used to describe motions. Recall John’s motion in Example 5: stays at B for 100 s 2 m s –1 for 500 s

P.45 Book 2 Section 2.1 Graphs of straight-line motion 3 Other motion graphs Distance-time graph Speed-time graph John’s distance-time and speed-time graphs: Cannot show the direction of motion!

P.46 Book 2 Section 2.1 Graphs of straight-line motion 3 Other motion graphs Example 6 Interpreting the v-t graph

P.47 Book 2 Section 2.1 Graphs of straight-line motion Example 6 Interpreting the v-t graph The v-t graph of a car: (a)What are the accelerations and displacements during 0–20 s, 20–50 s, 50–60 s and 60–65 s?

P.48 Book 2 Section 2.1 Graphs of straight-line motion Example 6 Interpreting the v-t graph 0–20 s: Acceleration = 15 – 0 20 – 0 = 0.75 m s –2 Displacement = = 150 m (20 – 0)  15  0.5

P.49 Book 2 Section 2.1 Graphs of straight-line motion Example 6 Interpreting the v-t graph 20–50 s: Acceleration = 0 50 – 20 = 0 Displacement = = 450 m (50 – 20)  15

P.50 Book 2 Section 2.1 Graphs of straight-line motion Example 6 Interpreting the v-t graph 50–60 s: Acceleration = 0 – – 50 = –1.5 m s –2 Displacement = = 75 m (60 – 50)  15  0.5

P.51 Book 2 Section 2.1 Graphs of straight-line motion Example 6 Interpreting the v-t graph 60–65 s: Acceleration = –7.5 – 0 65 – 60 = –1.5 m s –2 Displacement = = –18.75 m (65 – 60)  (–7.5)  0.5

P.52 Book 2 Section 2.1 Graphs of straight-line motion Example 6 Interpreting the v-t graph (b)Sketch the a-t graph of the car.

P.53 Book 2 Section 2.1 Graphs of straight-line motion Example 6 Interpreting the v-t graph (c) Describe the motion of the car in the journey. 0–20 s:accelerates uniformly from rest at 0.75 m s –2 20–50 s:at constant velocity of 15 m s –1 50–60 s:decelerates uniformly to a stop at 1.5 m s –2 60–65 s:moves in opposite direction and speeds up at 1.5 m s –2

P.54 Book 2 Section 2.1 Graphs of straight-line motion Example 6 Interpreting the v-t graph (d)Find the average velocity of the whole journey. Average velocity = total displacement total time taken = – – 0 = 10.1 m s –1

P.55 Book 2 Section 2.1 Graphs of straight-line motion Check-point 3 – Q1 Complete the motion graphs for each car. Car A

P.56 Book 2 Section 2.1 Graphs of straight-line motion Check-point 3 – Q1 Car B Complete the motion graphs for each car.

P.57 Book 2 Section 2.1 Graphs of straight-line motion Check-point 3 – Q1 Car C Complete the motion graphs for each car.

P.58 Book 2 Section 2.1 Graphs of straight-line motion Check-point 3 – Q2 A car moves as shown. Describe the motion of the car in different periods and draw the corresponding v-t and a-t graphs.

P.59 Book 2 Section 2.1 Graphs of straight-line motion Check-point 3 – Q2 0–2 s:It moves at a constant velocity of _________. 2–3 s:It makes a U-turn. Its velocity changes from __________ to _________. 3–5 s:It moves at a constant velocity of _________. 10 m s –1 –10 m s –1 10 m s –1

P.60 Book 2 Section 2.1 Graphs of straight-line motion Check-point 3 – Q2 v-t graph a-t graph

P.61 Book 2 Section 2.1 Graphs of straight-line motion 4 Motion analyzing tools a Data-logging system i Sensing motion Use motion sensors connected to a data-logger to record motions. The data-logger is connected to a computer which runs a data-logging program. Motion sensor

P.62 Book 2 Section 2.1 Graphs of straight-line motion i Sensing motion The motion sensor emits ultrasound signals and detects its echoes from a moving body. The data-logging program presents the collected data in s-t and v-t graphs. Data-logging: collection and processing of data Simulation 2.6 Studying motion using the motion sensor faster s i m p l e r

P.63 Book 2 Section 2.1 Graphs of straight-line motion i Sensing motion Expt 2a Studying motion using the motion sensor

P.64 Book 2 Section 2.1 Graphs of straight-line motion Experiment 2a Studying motion using the motion sensor 1 Set up the motion sensor as shown.

P.65 Book 2 Section 2.1 Graphs of straight-line motion Experiment 2a Studying motion using the motion sensor 2Facing the motion sensor, slowly move backwards. Note the s-t and v-t graphs on the computer screen. 3Move with different motions. Note the s-t and v-t graphs again. Video 2.1 Expt 2a – Studying motion using the motion sensor

P.66 Book 2 Section 2.1 Graphs of straight-line motion a Data-logging system Expt 2b Acceleration down a slope Simulation 2.7 Acceleration down a slope

P.67 Book 2 Section 2.1 Graphs of straight-line motion Experiment 2b Acceleration down a slope 1 Adjust the slope of the runway so that the trolley speeds up as it moves down. 2 Allow the trolley to move down. Note the s-t and v-t graphs obtained. runway trolley motion sensor

P.68 Book 2 Section 2.1 Graphs of straight-line motion Experiment 2b Acceleration down a slope An example of the result: s-t graph v-t graph Video 2.2 Expt 2b – Acceleration down a slope

P.69 Book 2 Section 2.1 Graphs of straight-line motion a Data-logging system ii Reading graphs Usually the direction away from the sensor is taken as +ve. Moving away from the sensor  velocity is +ve Example 7 Studying s-t graph from motion sensor  displacement becomes more +ve

P.70 Book 2 Section 2.1 Graphs of straight-line motion Example 7 Studying s-t graph from motion sensor The s-t graph of a girl is shown below. Describe the her motion. Linear Fit m (Slope) – b (Y Intercept) r Linear Fit m (Slope) – b (Y Intercept) – r

P.71 Book 2 Section 2.1 Graphs of straight-line motion Example 7 Studying s-t graph from motion sensor Initially, she is 2.2 m away from the sensor. Then she walks towards the sensor with an average velocity of 0.4 m s –1. When she is 0.2 m from the sensor, she walks away with an average velocity of 0.2 m s –1.

P.72 Book 2 Section 2.1 Graphs of straight-line motion b Motion video analysis We can also use software to analyze the motion on a video. Video 2.3 Video motion analysis (100 m splint) 4 Motion analyzing tools

P.73 Book 2 Section 2.1 Graphs of straight-line motion Example 8 Analyzing a motion video b Motion video analysis

P.74 Book 2 Section 2.1 Graphs of straight-line motion Example 8 Analyzing a motion video The s-t graph of an athlete in a 50-m race is shown below. First 3 points: form a curve Remaining points: form a straight line

P.75 Book 2 Section 2.1 Graphs of straight-line motion Example 8 Analyzing a motion video After fitting the s-t graph, we get the a-t graph as follows: s-t graph a-t graph

P.76 Book 2 Section 2.1 Graphs of straight-line motion Example 8 Analyzing a motion video (a)Describe the motion of the athlete during t = 0–2 s and t = 2–8 s. Accelerates during t = 0–2 s Constant velocity during t = 2–8 s

P.77 Book 2 Section 2.1 Graphs of straight-line motion Example 8 Analyzing a motion video (b)Estimate the velocity of the athlete at t = 2 s from the a-t graph. Acceleration at t = 2 s is 4.25 m s –2. Time / sVelocity / m s – His velocity at t = 2 s is 8.5 m s –1.

P.78 Book 2 Section 2.1 Graphs of straight-line motion Check-point 4 – Q1 The s-t graph of a trolley is shown below. What is the motion of the trolley from t = 1.0 s to 3.0 s?

P.79 Book 2 Section 2.1 Graphs of straight-line motion Check-point 4 – Q1 The trolley is moving with (decreasing/constant/increasing) velocity. It is moving (away from/towards) the motion sensor.

P.80 Book 2 Section 2.1 Graphs of straight-line motion Check-point 4 – Q2 The v-t graph of a trolley is shown below.

P.81 Book 2 Section 2.1 Graphs of straight-line motion Check-point 4 – Q2 Sketch the corresponding s-t graph. Assume the trolley is initially at 0 m.

P.82 Book 2 Section 2.1 Graphs of straight-line motion Check-point 4 – Q3 Which of the following is not a possible motion graph from a motion sensor? A B C

P.83 Book 2 Section 2.1 Graphs of straight-line motion Check-point 4 – Q4 Here is an s-t graph obtained by the motion video analysis software.

P.84 Book 2 Section 2.1 Graphs of straight-line motion Check-point 4 – Q4 What kind of motion does it describe? AMoving at constant velocity and then speeding up BSlowing down and then moving at constant velocity CSpeeding up and then slowing down DUniform motion

P.85 Book 2 Section 2.1 Graphs of straight-line motion The End