SUBJECT: PHYSICAL SCIENCES GRADE: 10 CHAPTER / MODULE: MECHANICS UNIT / LESSON TOPIC: - Equations of Motion - Graphs of Motion

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1 SUBJECT: PHYSICAL SCIENCES GRADE: 10 CHAPTER / MODULE: MECHANICS UNIT / LESSON TOPIC: - Equations of Motion - Graphs of Motion By the end of this unit, you should be able to: describe motion along a straight line. use graphs and mathematical formulae to answer questions about motion. answer ticker timer questions. know the difference between speed and velocity and distance and displacement. LESSON OVERVIEW (Knowledge areas) 1. Introduction 2. Position, Displacement and Distance 3. Speed and velocity 4. Acceleration 5. Description of motion in words, diagrams, graphs and equations 6. The ticker timer End of session 1 7. Equations of motion 8. Review questions and exercise 1 End of session 2 LESSON 1. INTRODUCTION Different types of movement are part of our daily life. People move due to the muscles that contract and relax, forcing the bones to move. Vehicles move due to the hydraulic systems that are built in it. In Science we describe the movements with our own special vocabulary. It is very important that you built your own personal dictionary to explain the meaning of different words and phrases in science. 2. POSITION, DISPLACEMENT AND DISTANCE Position: This is the place where one fines an object. The position of an object can only be described if there is a second point that can be used as a reference. One way of describing the position of an object is by referring to the compass directions (north, east, south and west).

2 (360 ) N (0 ) W (270 ) E (90 ) S (180 ) One always measure from north (0 ) clockwise, when the compass directions are used and only degrees are given. One measures 20 from east in the direction of south when the position is given as 20 S of E. Distance: Distance is the actual path followed from the reference to the new position of the object. Distance must always be measured in meters because it is the actual distance moved. Distance is a scalar because it only has a magnitude (size) because the direction can change all the time. House Shops Black: the distance you travel (by road) to the shops. Red: the displacement (the shortest distance from your house to the shops (as the crow flies) Displacement: It is the shortest distance from point A to be in a specific direction. Displacement will always have a magnitude (size) and direction and is called a vector. Vector: Scalar: Any quantity that has magnitude and direction. Any quantity that only has magnitude.

3 3. SPEED AND VELOCITY: Average speed The average speed is the total distance travelled divided by the time taken. Average speed is the rate at which the distance changes Speed is measured in m s -1 (meters per second). Average velocity The average velocity is the resultant change in position divided by the time taken. Average velocity is the rate of change of displacement. Velocity is measured in m s -1 with its direction the same as that of the change on position. Where: x is resultant distance t is the time taken v averaqge is the average speed The word rate implies that you divide by time. Where: x is resultant displacement t is the time taken v averaqge is the average velocity Instantaneous speed Instantaneous (momentary) speed is the speed at which an object moves at a particular moment in time (i. e. speed measured over an extremely short time interval) The speedometer of a car gives the momentary speed of the car. It has no direction and is called a scalar. Instantaneous velocity Instantaneous (momentary) velocity is the velocity at which an object is moving at a particular moment in time (i. e. velocity measure over an extremely short time interval. It has direction and is called a vector quantity. Acceleration: Acceleration is the rate (speed) at which velocity change. If the velocity increases at a specific rate in the same direction as the original movement, means that the acceleration is positive. If the velocity increases at a specific rate in the opposite direction as the original direction of movement, the acceleration will be negative and the object will slow down. To measure the acceleration the change in velocity over a specific time interval must be measured. The initial velocity (v i ) is the velocity before the change took place and the final velocity (v f ) is the velocity after the change took place.

4 4. DESCRIPTION OF MOTION IN WORDS, DIAGRAMS, GRAPHS AND EQUATIONS Motion can be described in different ways. Words: When your friend explains his first experience in driving a car and tells you in detail how he struggles to pull of and stop. Diagrams: When you draw a sketch to explain a specific movement. Graphs: We use three different graphs. 1. Velocity time graph 2. acceleration time graph 3. position time graph. 1. Velocity time graphs Every type of movement has a different line on the graph. Constant velocity Positive acceleration Negative acceleration v (m s -1 ) v (m s -1 ) v (m s -1 ) To calculate the displacement from a velocity time graph: calculate the area under the graph. Square: l x b (side x side) Rectangle: l x b Triangle: ½ b x h

5 To calculate the acceleration from a velocity time graph: Calculate the gradient of the graph. 2. Acceleration time graph Every type of movement has a different line on the graph. Non-uniform acceleration Positive constant acceleration Negative constant acceleration a (m s -2 ) a (m s -2 ) a (m s -2 ) To calculate the velocity from an acceleration time graph: calculate the area under the graph. Square: l x b (side x side) Rectangle: l x b Triangle: ½ b x h 3 Position time graph Every type of movement has a different line on the graph. Constant position Positive change in position Negative change in position x (m) x (m) x (m) To calculate the velocity from a position- time graph: Calculate the gradient of the graph.

6 5 TICKER TIMERS The period of the ticker timer The period of the ticker timer is the time between two successive dots on the ticker tape. It doesn t matter if the distance between the dots change, the period will still be constant How do we calculate the period? From the frequency! If the frequency of the ticker timer is 50 Hz, the period will be: The time between the dots are thus always known = the period of the ticker timer. Hint: Remember T is the symbol for period and t is the symbol for time. The velocity of a ticker timer How can one determine from the tape whether the trolley accelerated or moved at a constant velocity? If the dots are the same distance apart, the object is moving at constant velocity The time between the dots is constant because it is equal to the period! The distance between the dots is also constant. Thus v stays constant. If the distance between the dots increases, the object will move faster, thus positive acceleration. The time between the dots is constant because it is equal to the period! The displacement s between the dots increase. Thus v will also increase.

7 We can thus reason that if the displacement s between the dots decrease, the velocity will also decrease and there will be a negative acceleration. How does one determine average velocity form the tape? 10mm 30mm 50mm 70mm A B C D E The period of the ticker tape is 0,02 s The average velocity from A to B and the average velocity from B to C are calculated as follow: How does one calculate instantaneous velocity? (Remember:Instantaneous velocity is the velocity at a specific time or point.) It can not be calculated directly. The instantaneous velocity at point B (v B ) is equal to the average velocity from point A to C (v AC ) because B is in the middle between A and C. Not in the Middle in terms of the distance but in terms of the time! The acceleration of a ticker tape How does one determine the acceleration? 10mm 30mm 50mm 70mm A B C D E The period of the ticker tape is 0,02 s

8 The average velocity from A to B and the average velocity from B to C are calculated as follow: 10mm 30mm 50mm 70mm A B C D E V= 1m.s -1 v= 1,5m.s -1 t = 0,02 s V AB is the average velocity between A and B and it means that the velocity is in the middle between A and B thus it cuts the time in half!. The time between the 2 velocities are thus 0,01 + 0,01 = 0,02 s. END OF SESSION 1 6 SPEED, VELOCITY AND ACCELERATION Speed Speed is measured in m.s -1. Velocity Velocity is measured in m s -1. The direction of the velocity is always the same than the direction of the displacement. Remember: Speed and time is scalars, thus speed will also be a scalar! Remember: Displacement is a vector, thus velocity will also be a vector!

9 Acceleration Acceleration is the rate at which velocity change. (Rate means the speed over time). Acceleration is measured in m s -2. Acceleration is a vector. If the object moves faster the acceleration is in the same direction as movement. If the object slows down the acceleration will be in the opposite direction than the direction of movement. 7. EQUATIONS OF MOTION To be able to do calculation with equations of motion you need the following vocabulary: v f, v i, a and x are vectors direction must be given with any answer. v i : The direction of the object will be the same than the direction of the velocity. An object moving from rest means that v i = 0. An object moving with constant velocity means v i = v f. v f : The direction of the object will be the same than the direction of the velocity. An object that comes to a standstill means v f = 0. An object that comes to rest means v = 0. An object moving with constant velocity means v i = v f x: The direction of the straight line from the starting point to the end determines the direction of the displacement. a: If an object moves faster, a s direction will be in the same direction as the object. If an object slows down, a s direction will be opposite to the direction of the object. An object that comes to a standstill means a is negative. An object that comes to rest means a is negative An object moving with constant velocity means a = 0. t: Time is a scalar and can never be negative.

10 REVIEW QUESTIONS 1. Which one of the following is not a vector A. velocity B. acceleration C. time D. displacement 2. To calculate the velocity from a displacement time graph. A. calculate the area under the graph. B. Calculate the gradient of the graph C. Add all the values together. 3. Moving from rest means: A. final velocity is zero. B. Initial velocity is zero. C. No acceleration. 4. Which one of the following units is the same than Hz? A. s B. s -1 C. m D. m The definition of period is: A. The time taken between two dots. B. The number of dots per second. C. The distance between two dots. EXERCISES 1. Position, Displacement and Distance EXERCISE A: Draw the following vectors on the same axis. (remember to work to scale!) m in the direction m in the direction 60 N of W m North east EXERCISE B: A baby crawls for 5 m on a bearing 90 and then 3m on a bearing of 45. Determine 2.1. the total distance the baby crawled with the aid of a diagram, what the babies resultant displacement would be. 2. Speed and velocity EXERCISE: An athlete runs once around a 40 m track in 50 s. 1.1 Calculate the athlete s average speed. 1.2 Calculate the athlete s average velocity. 1.3 What is the athlete s displacement? Explain your answer.

11 3. Acceleration EXERCISE A: A car travel with a initial velocity of 40 m s -1 and then accelerates to 55 m s -1. It takes the car 15 seconds to reach the new velocity. Calculate the acceleration of the car. EXERCISE B: Thabo is cycling at 20 m s -1 and sees a whole in the road. He brakes and stops after 4 s. Caculate his acceleration. 4. Description of motion in words, diagrams, graphs and equations EXERCISE A: Draw the velocity time graph for the following description. A lift is acceleration from rest for 2 s and then move at a constant velocity for 5 s before it slows down to a standstill in 2 s. EXERCISE B: 2.1 Describe the motion of the train. v (m s -1 ) t(s) 2.2 Calculate the acceleration of the train between 20 and 28 s Calculate the distance travelled in the 44 s. 2.4 Draw a acceleration time graph from the velocity time graph. 5. The ticker timer EXERCISE A: The ticker tape shows the dots that are made by a ticker timer with a frequency of 50 Hz. A represents the first, B the 11 th, C the 21 st and D the 31 st dots. A B C D The distance between the respective dots is: AB = 12 mm, BC = 24mm, CD = 48mm 1.1 What is the period of the ticker timer? 1.2 What is the average velocity from A to B? 1.3 Calculate the average velocity from C to D. 1.4 Determine the acceleration of the trolley.

12 EXERCISE B: A car is accelerated from rest during a road test. The displacement values from the starting point for the first 8 seconds are given in the following table: Time (s) Displacement (m) 1 1,25 2 5,0 3 11, ,0 5 31, ,0 7 61, ,0 Calculate the average: 2.1 velocity during the third second. 2.2 velocity during the seventh second 2.3 acceleration between the third and the seventh second. 6. Equations of motion EXERCISE A: A car accelerates uniformly in 12 s from 10 m s -1 to a speed of 18 m s -1. Calculate the distance travelled by the car while it is accelerating. EXERCISE B: An aeroplane with a velocity of 45 m s -1 comes in to land at the start of the runway and brakes at -5 m s -1. Will it be able to stop in time if the runway is 275 m long? Use the necessary calculations to explain our answer.

13 HOMEWORK 1. An athlete runs from A to B around a circular track that has a radius of 100 m. (circumference of a circle is 2πr) A 100 m 1.1 Calculate the distance moved from A to B 1.2 Calculate the displacement from A to B. B 2. A motorcycle moving at 30 m s -1 due west on a straight road, brakes and comes to a standstill after 6 s. Calculate the acceleration of the motorcycle. 3. The graph is a position time graph for two motor cars K and L. At the instant t = 0 s the position of each car was marked x (m) K 16 8 L Identify the motor car with the greatest velocity How far apart are the two motor cars at t = 0 s? 4. Sipho is cycling at 5 m s -1 on a dirt road when he spots a bull 20 m ahead of him. It took him 30 seconds before he starts to apply his brakes. If he stops after 3 minutes, will he stops before he collides with the bull? END OF SESSION 2

2.1. Linear motion is a study of moving object in a line. We need a to describe and of objects.

2.1. Linear motion is a study of moving object in a line. We need a to describe and of objects. 2.1 Linear motion is a study of moving object in a line. We need a to describe and of objects. 34 Example of reference frames Note: Reference frame is presented by the coordinate system. We frequently