Kinematics Study of

Kinematics is the branch of physics that describes the motion of objects, but it is not interested in its causes.

Itziar Izurieta (2018 october) Index:

1. What is motion? ...... 1 1.1. Relativity of motion ...... 1 1.2.: Cartesian coordinate system ...... 1 1.3. and trajectory ...... 2 1.4.Travelled and ...... 3 2. Quantities of motion: and ...... 4 2.1. Average and instantaneous speed ...... 4 2.2. Average and instantaneous velocity ...... 7 3. Uniform linear motion ...... 9 3.1. Distance- graph ...... 10 3.2. Velocity-time graph ...... 11 4. Velocity change: ...... 14 5. Uniformly accelerated linear motion (UALM) ...... 16 5.1. Graphs in uniformly accelerated motion ...... 17 5.2. Vertical motion due to : Free fall...... 19 Common mistakes and misconceptions ...... 20 Vocabulary ...... 31

1- What is motion? 1.1- Relativity of motion 1.2- Position and trajectory 1.3- Displacement and travelled distance 2- Speed 2.1- Average speed and instantaneous speed. 2.2- Average velocity and instantaneous velocity. 2.3- Uniform linear motion: graphs and equations 3- Velocity change 3.1- Acceleration. 3.2- Uniformly accelerated linear motion 3.3- .

1. What is motion? Even when we are standing still on the Earth we are actually travelling at extremely high . This is because the Earth is constantly spinning (rotating) on its axis and travelling around the Sun. Even the Sun isn’t standing still; it is travelling at an incredibly high speed around our galaxy. In physics, motion is the study of how something moves, whether it is a planet moving around the Sun or a snowboarder flying through the air. An object’s motion is usually described in terms of its velocity and acceleration, and its motion only changes if a or several act on it.

1.1. Relativity of motion How can we know, in an objective way, what is in motion? And what is stationary? How can we know if something is moving or still? Motion is the change in position of an object with respect to another one: it is the change of position that an object goes through as time goes by relative to a reference system that we take as still and fixed (relative to a fixed reference that we take as stationary).

Taking this definition into account motion is relative. Each object in the universe is moving relative to the rest. An object may appear to have one motion to one observer and a different motion to a observer. For example, if we are travelling in a car, we are still relative to the car (if the car is our reference system), but we are in motion relative to the road or to any tree at one side of the road (if the road or the tree are our system of reference). So, motion is a relative concept, we measure it relative to an object or an observer that we consider not to be moving. We describe motion in terms of position, time, displacement, velocity and acceleration. A body that does not move is said to be at rest, motionless, immobile, stationaryor to have a constant time- invariant position. ** In order to simplify the study of motion and unless otherwise stated, we consider every object in the universe as a point, known as a material point.

1.2. Frame of reference: Cartesian coordinate system The reference system is the frame of reference, also called reference frame. A framework that is used for the observation and mathematical description of physical phenomena and the formulation of physical laws, usually consisting of an observer, a coordinate system, and a clock or clocks assigning at positions with respect to the coordinate system. In a Cartesian coordinate system, the way to locate objects is based on a set of perpendicular axes that intersect at the orgigin. To locate an object in we need a frame of reference with three axes. But, this year, however, we will study movemente in a plane and two axes will be enough.

Three Two dimension

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1.3. Position and trajectory Trajectory (or path): this is the path that a moving object follows through space over time. A trajectory may be linear (straight) or curved. This is obtained by joining all the positions that an object takes in its motion. Position: this is the point where an object is located at any relative to a system of reference. For example, if we use a road as a system of reference and any point, O, in this road as the origin, we know the position of a car if we know the distance till the origin.

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There are many types of motion:

Depending on the shape of the path, a given motion is classified:  Linear when it is a straight line  Curvilinear if it is not a straight line. In the special case of movement along a circumference, we call a circular motion.

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This year we will look more closely at linear motion and circular motion, fron which , in the future, you will be able to study more complex motion.

1.4. Travelled distance and displacement . Travelled distance, s ( is a greek letter to express change), is the length of the path the object or body has described:

s = s - so If the path is straight and parallel to the X axis, we use x to express the distance travelled. If the path is straight and parallel to the Y axis, we use y to express the distance travelled. In other cases (round, elliptic paths...) we use s to express the distance travelled.

. Displacement is a vector quantity. Displacement is distance in a given direction. It is a vector quantity (it has a direction associated with it). To define a vector quantity it is necessary to use its size (value), its units, its direction and sometimes also its origin. It is the vector from the starting point to the end point. It is obtained by making the subtraction of the final and the initial position. If we use only one of the axis, we don’t need to use the vector expression and the signal of the subtraction will give us information about the direction of the motion in the same way. Displacement from the initial position (1,3) m to (4,2) m. The curved line is thetrajectory between these two points

If trajectory is straight  travelled distance = displacement

The SI unit of distance and displacement is the meter [m].

-2 -1 0 1 2 3 4 5 6 7

a) From 1 m to 5 m: x = 5-1 = 4 m to the right  Displacement = Travelled distance b) From 5 m to 6 m: x = 6-5 = 1 m to the right  Displacement = Travelled distance c) From 6 m to -1 m: x = -1 -6 = -7 m to the left  |Displacement| = Travelled distance d) From -1 m to – 6 m: x = 6-(-1) = 7 m to the left  |Displacement| = Travelled distance

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Activities:

1. A cyclist has travelled around a semicircular curve of 50 m in radius. How much distance, s, has he travelled? And what displacement, x, has he travelled? Do they have the same value? Why? Draw it

2. The initial position of a moving object is -400 m and the final one is 13 km. Knowing that its path was straight, what distance has he travelled? Draw the displacement vector.

3. Give an example of an object that has a displacement, x, of 0 after having moved.

4. A tank travels 12 km to the North, then it turns East and travels 5 km. What is the total distance travelled by the tank? What is the displacement of the tank?

5. A toy train moves along a circular path of radius 20 cm as shown in the picture. What is the distance and displacement travelled when the train a) moves from O to P? b) moves from O to and then back to O?

2. Quantities of motion: Speed and velocity 2.1. Average and instantaneous speed Cars have speedometers. But how can you tell how fast you are going on your bike? Most bicycles do not have speedometers, so you cannot measure your speed directly. But you can time how long it takes to cycle between two places, two lamp-posts, for example. And you can measure how far you have cycled: the distance between the two lamp-posts. Then you can out the average speed. Speed is the rate of change of distance with respect to time. The symbol for speed is v. Speed is a scalar quantity. Its unit is the meter per second (m.s-1 or m/s). It can be measured in kilometres per hour (miles per hour in England!). An object has a constant speed if it neither speeds up nor slows down. An object moves with constant speed if its average speed is always the same, no matter over which part of the journey it is measured. The words steady or uniform are sometimes used instead of constant. Most moving objects that we see around us have varying speed. Sometimes they are speeding up, at other times they are slowing down. For example, if you drop something, it moves downwards with its speed increasing.

Average speed is a scalar quantity. The average speed in an interval of time is the distance travelled divided by the duration of this interval |푆푓 − 푆𝑖| |∆푠| 푣 = = 퐷𝑖푠푡푎푛푐푒 푡푟푎푣푒푙푙푒푑 푚 푡푓 − 푡𝑖 ∆푡 퐴푣푒푟푎푔푒 푠푝푒푒푑 = 푡𝑖푚푒 푡푎푘푒푛 x (horizontal motion) or y (vertical motion) Speed has the of a length divided by a time, so the SI Units are meters per second, m/s.

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Activities:

6. A sprinter runs 200 meters in 22 . Calculate her average speed over the 200 m 9,09 m/s

7. A man walks with an average speed of 2 m·s-1. How long will it take him to walk 1 km? 500s

8. A dog runs along a road at a constant speed of 3 m/s. a) How far will it travel in 10 s? b) How far will it travel in ¼ hour? 30m; 2,7·103m 9. A car has a steady speed of 63 km/h. a) How far does it travel in 12 seconds? b) How long does it take it to travel 400 m? 210 m; 22,86 s 10. Find the average speed in meters per second of a car that travels: a) 2000 m in 4 minutes. b) 100 m in 10 seconds. 8,33 m/s; 10 m/s 11. A woman travels a distance of 200 km with an average speed of 25 m/s. How long does the journey take? 2,22 h; 8.103 s 12. If the average speed of a plane over a 2000 km journey is 200 m/s, how long does the journey take? 1.105s 13. A boy walks once around a circle of radius 30 m in 1 minute 30 seconds. What is the average speed for the journey? 2,1 m/s 14. A girl walks along a road at a constant speed of 2 m/s. How far will she travel in:

a) 25 b) ½ hour c) 1 ms d) t seconds

15. The fastest land mammal (the cheetah) and the fastest fish (the sailfish) have the same highest recorded speed of 110 km/h. At top speed, how far would a sailfish travel in an hour? 110 km 16. An athlete runs 100 m in 10 s. What is the speed of the athlete? And what is the speed of a car that travels 200 km in 4 hours? 10 m/s; 50 km/h; 13,89 m/s

17. Copy and complete the below Speed (m/s) Distance (m) Time (s) 100 5 45 9 40 20 20 2 6 3

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18. Work out how far the following people will travel: (answer in m) a) A man walking at 3,5 kilometres per hour for 3 hours b) A student running at 5 meters per second for 10 seconds. c) A baby crawling at 0,1meters per second for half a minute. 1,05·104 m; 50 m; 3 m However, knowing the average speed is sometimes not very useful. For most journeys, the speed is not always the same. It varies. For instance, imagine you are going to drive to a friend’s house: 1- Town traffic. 3- Country lane. You turn off the motorway. During the first You have 10 more part of your kilometers to go. journey, you drive But you get held from home to the up on a narrow motorway. This country road. It takes you through busy city streets. The takes you 30 speed limit is 50 kmph, but often you are minutes (0,5 hours). So your average travelling slower than this. speed on this part of the journey is 10 푘푚 = 20 푘푚/ℎ 2- Motorway. You 0,5 ℎ travel on the 4- You look at your watch and the odometer motorway for 1 when you arrive. The whole journey of hour. In that 120 kilometers has taken you exactly 2 time, you go 100 hours. So, for the whole journey, your kilometers. So 120 푘푚 average speed was = 60 푘푚/ℎ your average speed on the motorway 2 ℎ 100 푘푚 is = 100 푘푚/ℎ 1 ℎ

So, you may know the average speed for the whole journey. But you cannot tell anything about how fast the car was going at any particular moment. If you did drive steadily at 60 km/h (average speed) for the whole journey, it would take you exactly 2 hours. But in practice, you did not. Your speed kept changing.

Activities: 19. Your whole journey above was 120 km. a) How far was it from home to the motorway? b) Altogether, you drove for 2 hours. How long did it take you to reach the motorway? c) So, what was your average speed in km/h from home to the motorway? 10km; 0,5h; 20 km/h; 72 m/s 20. The movement of an object is given by the following graph. a) Explain her motion in intervals a, b, c and d b) Find the total distance travelled and the speed on each interval c) Calculate the average speed

10m; 0,4 m/s; 0 m/s;0,4 m/s; 1,2 m/s; O,5 m/s

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The speed at a particular moment is called the instantaneous speed. If you were able to calculate average speeds over shorter and shorter time intervals, these would get closer and closer to the instantaneous speed, you measure the average speed over a very short time interval.

The speedometer in a car shows the driver the instantaneous speed.

For example, in 1999 in the men’s world record for the 100m race was 9,84 seconds. This is an average speed of 10,16 m/s. After running 1 m of the race, though, the winner’s instantaneous speed might have been 3 m/s. After a few more meters, it could have been as high as 11 m/s.

Speed is a scalar quantity. This means that it measures the amount of speed a moving object has, but not the direction in which it is moving. In Physics, the unit of measurement most often used is meters per second (m/s). However, you will often see kilometers per hour (km/h) as well. This measures how far something will go in an hour.

Activities: 21. A motorcyclist has travelled 500 m in half a minute. What is his average speed? Is it possible to have an instantaneous speed higher than the average speed when time is t=25s?

2.2. Average and instantaneous velocity An object can move from one position to another with greater or lesser speed and, if its path is curved, the direction of travel will always change. Velocity accounts for all of that.

Velocity is the that describes how a moving object´s position changes.

It is a vector quantity since it must convey both: the speed and the direction of motion.

 Average velocity, 푣⃗⃗⃗푚⃗⃗ : ∆푟 . The vector: 푣⃗⃗⃗⃗⃗ = 푚 ∆푡

∆r . Its magnitude: v = m ∆t

Velocity is the rate of change of displacement with respect to time. The usual symbol for velocity is v. Velocity is a vector quantity For example: 5 m/s way: horizontal; direction: to the right; module: 5; units: m/s

Although this car is travelling at a speed of 20 m/s (steady), its velocity is changing all the time because it is constantly changing direction.

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Velocity is, also, measured in m/s in a particular direction. For instance, someone walking east at 1,5 m/s has a velocity of 1,5 m/s.  If she is walking west? Should it be 1,5 m/s too?

 Instantaneous velocity, 푣 : If we calculate the average velocity for a small enough time interval (the mathematical expression is t → 0) we obtain the velocity at a given instant, called the instantaneous velocity. Instantaneous velocity, 푣 , specifies the velocity of an object an each instant in time.

If a body has constant velocity it does not speed up, slow down or change direction. Its average velocity is the same no matter what time interval during the motion it is measured.

Activities:

22. Two trains are travelling at 100 km/h. One train is travelling north and the other south. Compare their .

23. The Olympic games were held in the UK in 1948. Here are some of the results:  Melvin Patton ran 200 meters in 21,1 seconds.  Dorothy Manly had a mean (average) speed of 8,2 m/s in the 100 m  Curtis Stone completed 5 kilometers in 14 minutes and 39,4 seconds  Henry Eriksson ran the 1500 meters in 3 minutes and 39,4 seconds. a) Which athlete ran the fastest? b) If all the athletes ran at a constant speed, only one would have had a constant velocity. Which one? Explain your answer. c) Find out how fast the winner of the woman’s 100 m sprint was at the last Olympics. Curtis Stone; Dorothy

24. In 12 seconds a girl travels from A to B by the path shown in the figure. The total distance travelled is 68 m. The overall displacement she undergoes is 40 m North East. Calculate:

a- Her average speed b- Her average velocity for the journey

25. In the next figure what is the displacement of: a- A from B b- B from C c- B from A d- C from B e- A from C f- C from A A

2 m

B C 2 m

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26. Use the to determine the average speed and the average velocity of the skier during these three minutes

3. Uniform linear motion (ULM) It is also known as uniform rectilinear motion or uniform straight motion (USM).

 Uniform means that the speed is constant, so the magnitude of the velocity doesn’t change.  Linear (straight) means that the trajectory is a straight line,

so the direction and sense of the velocity is constant.

If the magnitude, direction and sense of the velocity do no change with time, we conclude that:

A uniform linear motion means that it has constant velocity (meaning that the speed, direction and sense do not change over time).

If we begin to count time in the very moment the object begins to move, then to=0 and the equation becomes: x = xo + v·t Where:  x is the final position (m)  x0 is the initial position of the object (m)  v is the speed (m/s)  t is the time (s) Characteristics of uniform linear motion:  It covers equal distance in equal interval of time.  Distance is directly proportional to time.  Average velocity and instantaneous velocity are the same.

 In this equation of motion x = 20 + 5t, what is the initial position? What is the speed?

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Activities:

27. Where will an object be after 25 minutes if its position at start is 3 km and its speed 90 km/h? What distance has it travelled during this time? Deduce the equation of the motion.

28. Using this graph find: a) The velocity of each object in m/s b) The equation of each motion (use the units of the graph) c) The distance between the two objects at 20 minutes.

3.1. Distance-time graph Distance-time graphs show how far an object has moved at different times. Time is plotted on the x-axis and distance is plotted on the y-axis. The speed can be calculated by finding the gradient (steepness, slope) of the line. This is done by dividing the change on the y-axis by the change on the x-axis. The distance-time graph of a uniform linear motion is a straight line which can passes through the origin or not.

Examples of some distance-time graph. Think about the or value of different magnitudes.

xo =

v =

xo =

v =

xo =

v =

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This graph shows the displacement-time graph of a car. From time t= 0 to t = 5 s, the car moves forwards, and at t= 5 s it has a displacement of 60 m. Then it remains stationary there for 5 s, and finally moves back to its starting position in another 5 s.

Calculate the velocity of each interval.

3.2. Velocity-time graph The velocity-time graph of a uniform linear motion is a horizontal line. a) If the mobile goes towards the right

Here the velocity is constant and positive, so the object is moving to the right or upwards.

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b) If the mobile goes towards the left side

The velocity is constant and negative, so the object is moving …………………………. or ……………………….

Activities:

29. Draw distance-time graph for the following journeys: a) A girl runs at a steady speed to a bus stop, where she suddenly stops. She gets on a bus. The bus travels at a steady speed that is faster than the girl’s running speed. b) A driver parks the car on a hill but forgets to leave the hand-brake on. The car starts to move and speeds up until it reaches a steady speed. Unfortunately there is a river at the bottom of the hill which the car rolls into. The car slows down and falls at a slower, steady speed in the water until it hits the bottom where it stops.

30. Next graph shows the displacement-time graph of a car during parking. Describe the motion of the car and find the velocity in each time interval.

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31. A boy walks at a velocity of 0, 5 m/s along a street for 50 s, suddenly he remembers that he has to buy something in a shop that he has passed by, so he turns around and walks at a velocity of -1 m/s for 10 s. He then stops for 30 s at the shop, and finally walks forward again at 0, 5 m/s for another 20 s. a displacement-time graph for the motion of the boy. 32. Some of these s-t graphs are wrong. Can you explain why? If they are right, describe the motion shown.

33. A man rides out into the country at a uniform rate of 30 km per hour. He rests 2 hours and then rides back at 20 km per hour. It took him 5 h to get back to the initial point. How far did he go? 36 km 34. Two trains start from the same station and run in opposite directions. One runs at an average rate of 40 km/h and the other at 65 km/h. How long will it take them to be 315 km apart? 3h 35. Two automobiles start from the same place and travel in opposite directions, one averaging 45 km per hour and the other 30 km per hour. How long will it take them to be 900 km apart? Draw the graphs of both . 12 h 36. Two young people, Aitor and Bea, start towards each other at the same time from points 510 km apart. If they travel 40 and 45 km an hour respectively, how long will it take them to meet? 6 h 37. Jones and Brown start from two points, which are 375 km apart and travel toward each other. The latter (the last) travels twice as fast as the former (“aurrekoa”). They meet in 5 hours. Find the rates per hour. Draw the graphs of both motions 25 km/h; -50 km/h 38. A motorboat starts out and travels 9 km an hour. In 3 hours another motorboat traveling 18 km an hour starts out to overtake the first one. How long will it take the second boat to overtake the first one? 6 h; 54 km 39. A freight train is traveling 30 km per hour. An automobile starts out from the same place 1 hour later and overtakes the train in 3 hours. What was the speed of the automobile? 40 km/h Kinematics (I.I) 13

4. Velocity change: Acceleration The fact that we can describe a motion´s average speed does not mean that the speed is the same throughout its path. Indeed, if the path is curved, the direction of the velocity is always changing. But also we can change our speed our motion. Acceleration is the physical quantity that describres changes in the velocity of an object.

An object is said to be accelerating if its velocity is changing in any way. Thus an object is accelerating if it is: . Speeding up . Slowing down . Changing direction . Speeding up as it changes direction . Slowing down as it changes direction Acceleration is the rate of change of velocity with respect to time. Acceleration is a vector quantity since it has a direction. The symbol for acceleration is a. Average acceleration is defined as follows: 푐ℎ푎푛푔푒 𝑖푛 푣푒푙표푐𝑖푡푦 푓𝑖푛푎푙 푣푒푙표푐𝑖푡푦 − 𝑖푛𝑖푡𝑖푎푙 푣푒푙표푐𝑖푡푦 퐴푣푒푟푎푔푒 푎푐푐푒푙푒푟푎푡𝑖표푛 = = 푡𝑖푚푒 푡푎푘푒푛 푡𝑖푚푒 푡푎푘푒푛

푚/푠 Since the unit of velocity are m/s and unit of time s: 푢푛𝑖푡 표푓 푎푐푐푒푙푒푟푎푡𝑖표푛 = = 풎/풔ퟐ 푠

The acceleration of an object tells us how quickly its velocity is changing. The more the velocity of an object changes is a certain time, the greater the acceleration. A negative acceleration is sometimes called deceleration.

https://es.khanacademy.org

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 What is the acelaration a<0 or a>0?  If an object is speeding up and moving to the right, a<0 or a>0?  If an object is speeding up and moving to the left, a<0 or a>0 ?  If an object is slowing down and moving to the right, a<0 or a>0?  If an object is slowing down and moving to the left, a<0 or a>0?

Activities: 40. Work out the acceleration of each of these cars: a) A car that gains (obtains) a speed of 20 m/s in 10 seconds b) One that gains 20 m/s en 5 seconds c) One that gains 30 m/s in 15 seconds d) One that gains 10 m/s in 2 seconds. e) Which car has the largest acceleration?

41. A car driver going at 90 km/h sees an obstacle on the road, pushes the brake and stops its vehicle in 5 s. a) What has its acceleration of braking been? b) What is the meaning of the minus sign of the previous result? -5 m/s2 42. At take-off, a plane accelerates from rest to 40 m/s in 10 seconds. What is the acceleration of the plane? 4 m/s2 43. In a car’s magazine we have found the next information about small utility cars: A model B model C model D model Largest speed (km/h) 150 155 150 151 Time(s) from 0 to 100 km/h 13,8 15,4 13,4 17,4 Taking these data into account, calculate: a) The acceleration of the four models in m/s2 2 m/s2; 4 m/s2; 2 m/s2; 5 m/s2 b) The time each one of them will need for getting the largest speed being still at the beginning. 20,7s; 23,8 s; 20,1 s; 26,26 s

44. Near the end of a race, a sprinter accelerates. She was running at 7 m/s but accelerates to 9 m/s in 4 seconds. What is the acceleration of the sprinter? 0,5 m/s2 45. A car is going round a roundabout at a steady speed of 30 km/h. Explain why it is accelerating, even though its speed is not changing. 46. Observing the graph, calculate the acceleration that a moving object has at each time interval.

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5. Uniformly accelerated linear motion (UALM) It is also known as uniformly accelerated rectilinear motion. (or straight: U.A.S.M.)  Uniformly accelerated means that the magnitude of the acceleration is constant, so the magnitude of the velocity changes at the same rate.  Linear means that the trajectory is a straight line, so the direction and sense of the acceleration is constant. If the magnitude, direction and sense of the acceleration do no change with time, we conclude that:

A uniformly accelerated linear motion means that the acceleration is constant (meaning that the magnitude, direction and sense of the acceleration do not change over time).

Since velocity is not constant, we have two equations: . Velocity time equation:

. position-time equation:

Activities: 47. A car changes its velocity from 0 to 100 km/h in 12 seconds. What is its acceleration? 2,3 m/s2 48. A car moving at 1,5 m/s accelerates at 0,2 m/s2 when it passes by x = -2 m. What is its direction? Is it an accelerated or decelerated motion? Calculate its position after 10 seconds have passed, the velocity in that moment and the travelled distance till then. 23 m; 3,5 m/s; 25 m 49. The velocity-time equation for an object is the following one: v = -3 – 0,8 t. a) Explain the characteristics of this motion b) If the object has started from x=3m, write the position-time equation and calculate the travelled distance in 6 seconds. 29,4 m 50. A vehicle reaches a speed of 72 km/h in 20 s beginning at rest. a) Work out its acceleration b) Calculate the travelled distance in that time 1 m/s2; 200 m

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5.1. Graphs in uniformly accelerated motion

Position-time graph: The position-time graph in a uniformly accelerated linear motion is a parabola that does not necessarily have to go through the origin.

This object is going: This object is :  Faster because the slope is increasing  Going slowing down, because the slope is decreasing  Backward  Forward

 Draw this two position- time graphs: The object is going faster and forward The object is going slowing down and backward

Velocity time graph: The velocity-time graph of a uniformly accelerated linear motion is a straight inclined line.

The velocity is increasing uniformly. The velocity is decreasing uniformly.

Acceleration-time graph: The acceleration is the rate of velocity change over time. In a uniformly accelerated linear motion, it is constant so it is a straight line. It aslo can be negative acceleration straight line.

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 Match these s-t and v-t graphs, and explain them (forward or backward; speeding up or slowing down; a<0 or a>)

Activities: 51. An object moving at 20 m/s breaks at a rate of 3 m/s2. How long does it take to stop completely? What distance has the object travelled during this time? t = 6,7 s; x = 66,6 m 52. (Activity FOR GROUP) Analyse this 3 graphs for the same motion of an object for 30 seconds. Explain the movement the object has made during this period.

http://hyperphysics.phy-astr.gsu.edu

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5.2. Vertical motion due to gravity: Free fall. Naturally, objects fall. This is due to gravity of the earth. In all what it will follow, we assume that air resistence has a negligible effect on a falling object, so that the approximation of the object's acceleration is due entirely to gravity is valid. In practice, there is usually air resistance. (1564-1642) made quantitative studies to find that the acceleration due to gravity is constant during the fall. Further, this acceleration is the same for any falling object. The magnitude of this accelation is represented by the symbol g and has the value of 9.8 m/s2.  How much must be the acceleration on the Moon? Greater or smaller than on the Earth?

 It is a type of uniformly accelerated rectilinear motion UARM . motion along a straight line . with variable velocity . constant acceleration because of the gravity and it is always 9.8 m/s2  It can be described with three equations:

Where:  h (or y) is the final position (m)

 h0 (yo) is the initial position of the object (m)

 v0 is the initial speed (m/s). It can be positive if the object is vertically thrown up in the air, zero if the object is dropped and negative if the object is vertically thrown down.  v is the speed in other moment (m/s)  t is the time (s)  g is the acceleration of gravit. Notes:  We will set the origin of motion on the Earth’s surface, so the gravity will be always negative.  When an object falls freely its velocity is negative (downwards) and it increases uniformly, so it accelerates.  When moving upwards, velocity is positive, but it decreases uniformly, so it decelerates.). In order not to make mistakes, we will take that into account in the equations).

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A Ball thrown vertically: When a ball is thrown up in the air, the ball's velocity is initially upward. Since gravity pulls the object toward the earth with a constant acceleration g, the magnitude of velocity decreases as the ball approaches (reaches or arrives at) maximum height.

At the highest point in its trajectory, the ball has zero velocity, and the magnitude of velocity increases again as the ball falls back toward the earth.

Direction of velocity and acceleration for a ball thrown up in the air.  Common mistakes and misconceptions: People mistakenly (by mistake) think the final velocity for a falling object is zero because objects stop once they hit the ground. In physics problems, the final velocity is the speed just before touching the ground. Once it touches the ground, the object is no longer in freefall.

Activities: 53. A stone has been thrown upwards with an initial velocity of 60 m/s. What will its height be after 5 seconds? How long will it take to get the highest position? 177,5 m; 6,12 s 54. We have released (freed) an object from a certain height. If it has arrived to the ground at a speed of 49 m/s. Calculate the time needed for freely falling and the height from where it has fallen. 5s; 122,5 m 55. A brick falls from a building under construction and at a height of 30 m. Calculate: a) The time it takes to reach the ground b) The velocity at that moment

2,47 s; -24,2 m/s

56. A ball falls freely from a window at 20 m from the ground. When will it reach the floor? What is its final velocity? 2,02 s; -19,8 m/s

57. An object falling from a fourth floor needs 2 seconds to reach to the ground. What is the height of the fourth floor? 19,6 m 58. A body of 100 kg falls freely from a height of 100 m. What is its acceleration after 1 second? And after 3 seconds? What about its velocities? - 9,8 m/s; -29,4 m/s

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6. Some interesting graphs:  Position-time graphs:

 Uniformily accelerated motion graphs (resolution of activity nº 52):

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7. Circular motion In Physics, circular motion is the movement of an object along the circumference of a circle. Examples: An artificial satellite orbiting the Earth. A stone which is tied to a rope and is being swung in circles. A car turning through a curve in a race track. An electron moving perpendicular to a uniform magnetic field. A gear turning inside a mechanism. Examples of Circular motion: https://www.youtube.com/watch?v=_ZYlel2Bs8g A circular motion can be positive or negative:

 It´s negative (-) if it is clockwise  It is positive (+) if it is anti- direction clockwise and

To express this motion we can use linear quantities/magnitudes or angular quantities/magnitudes. Linear magnitude s v a Angular magnitude   

7.1. Unifor circular motion (UCM) If the speed of the object remains the same then it is uniform circular motion. However the velocity does change, as the direction of the velocity is continually changing (though not its module). This means the object is constantly accelerating towards the centre, so there is a force acting towards the centre. This is called the .  :  When a body moves along an arc of a circle, of radius r, between points a and b, the that the radius sweeps out at the centre is called the angular displacement.

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Specifications:  are measured in degrees or . Equivalence: 2휋푟 360° = = 2휋 푟푎푑 푟  The angle has sign positive (+) if it is anti-clockwise and it´s negative (-) if it is clockwise direction.  Relation between angular displacement and linear displacement: 푠 휃 = (휃 𝑖푛 푟푎푑𝑖푎푛푠) 푟 r: radio of the circle

 Convert 180º to radians:

 Convert 30º to radians:

 Convert  radians to degrees:

 Angular speed or velocity: ω is the rate of change of angular displacement and can be described by the relationship:

and as ω is constant, the angle can be calculated from:

 Calculate angular speed of Earth:

 Period: T It is the time for an object to complete one revolution. SI units are seconds. As the distance travelled in one revolution is: 2πr 2πr 2πr v = → T = T v  : f It is the number of revolutions the object will complete in one second; it´s inversely proportional to period. It is measured in units of s-1, also called Hz (): 1 푓 = 푇

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Take notice:  Angular speed and linear speed: ω We know that linear speed: ∆푠 v = ∆푡 where ∆s is linear displacement of arc, and 푠 휃 = 푟

Thus, linear speed: ∆푠 r· ∆θ v = = → v = r · ω ∆푡 ∆t Hence, angular speed: v ω = r Where v is equivalent to the linear speed. This is the relation amongst angular speed, linear speed, and radius of the circular path.

From this relation, one can compute this speed. Linear velocity is also sometimes called tangential velocity. Tangential velocity is the straight line velocity of a point on the edge of a rotating object. It is called tangential velocity because the direction of motion is tangent to the circular path. It is the path we could imagine an object taking if we had it on a string and rotating it, but then we let it go. Simulation: https://conteni2.educarex.es/mats/14359/contenido/

Angular velocity does not change with radius, but linear velocity does.

For example, in a music band line going around a corner, the person on the outside has to take the largest steps to keep in line with everyone else.

Therefore, the outside person has a much larger linear velocity than the person closest to the inside. However, the angular velocity of every person in the line is the same because they are moving the same angle in the same amount of time.

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 Centripetal acceleration: ac Can an object accelerate if it's moving with constant speed? Yup! Many people find this counter-intuitive at first because they forget that changes in the direction of motion of an object—even if the object is maintaining a constant speed—still count as acceleration. Acceleration is a change in velocity, either in its magnitude—i.e., speed—or in its direction, or both. In uniform circular motion, the direction of the velocity changes constantly, so there is always an associated acceleration, even though the speed might be constant. You experience this acceleration yourself when you turn a corner in your car—if you hold the wheel steady during a turn and move at constant speed, you are in uniform circular motion. What you notice is a sideways acceleration because you and the car are changing direction. The sharper the curve and the greater your speed, the more noticeable this acceleration will become.

This acceleration es called the centripetal force or radial acceleration. The centripetal acceleration is pointed towards the center of a curved path and perpendicular to the object’s velocity. Causes an object to change its direction and not its speed along a circular pathway. v2 a = = r · ω2 c r SI units are m/s2

. This animation shows centripetal acceleration as a limit of successive "bounces" off of a wall, each bounce gives a push towards the center of the circle. https://www.youtube.com/watch?v=fSfVVz0eIis

. In this animated physics video, your students will learn about centripetal force and 's second law. https://www.youtube.com/watch?v=KvCezk9DJfk

Activities: 59. How many degrees are  rad, /2 rad and /4 rad? How many radians are 2 cycles? 60. Have look at the carousel and answer these questions: a) Where is the horse at the beginning? b) Where is the horse after 5 seconds? And at 10 s? c) Where is the horse again after 20 s from the start of the motion? d) What is the displacement of the horse after 20 s? t=10s t=0 s; t=20s e) How many radians has the horse moved through in each case? f) How long does it take the horse to complete a whole cycle?

g) What is the angular speed of this motion? Kinematics (I.I) 25

T=20s T=20s 61. Express the following angular speeds in radians per second: 1 cps and 1 cpm 2 rad/s; 0,033 rad/s 62. Calculate the angular speed and the linear speed of the Moon. It is known that the Moon needs 28 days to complete a whole cycle and that the distance from it to the Earth is 3,84·105 km. 8,27·10-7  rad/s; 997 m/s

63. A wheel of 20 cm in diameter turns round at 60 cpm. Calculate: a) The angular speed of the wheel b) The linear speed at a point in the periphery of the wheel c) The frequency and period time of the wheel 2 rad/s; 1,25 m/s; 1 Hz 64. A wheel in a car has a radius of 40 cm and it completes 700 cycles in 10 minutes. Calculate: a) The angular speed in radians per second b) The linear speed of a point of the periphery of the wheel c) The angular speed and linear speed of a point close to the centre (1 cm) d) The number of cycles completed in 2 hours e) The distance travelled in 15 minutes f) The time necessary for travelling a distance of 1 km g) The time necessary for travelling an angle of 3 radians. 2,33 rad/s; 2,9 m/s; 7,32 .10-2 m/s; 8400 cycles; 2,639 .103 m; 3,41.102 s; 1,28 s 65. A fan’s wings turn round at an angular speed of 600 cpm and linear speed of 16 m/s. Work out: a) The radius of the wings b) The period time of the fan 0,25 m; 0,1 s

66. A motorcycle wheel spins at 1000 rpm. If its radius measures 50 cm, calculate: a) The angular speed in SI units b) The linear speed c) The period and the frequency; d) Its number of complete turns in 15 s 104,7 rad/s; 52,3m/s; T=0,03s f=33.3 Hz; 250 turns 67. A wheel with a radius of 15 cm goes round three times in 0.5 seconds. Calculate: a) Its angular velocity b) Its centripetal acceleration c) Its period and frequency. d) The number of times the wheel would go round in 2 minutes. a)37,6 rad/s b) 213.2 m/s2 c)0.167 s 6 Hz d)720 cycles

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KINEMATICS: Final exercises

1. Write the meaning of these concepts in a short way: a- System (frame) of reference e- Velocity i- Angle b- Path (trajectory) f- Uniform motion j- c- Travelled distance g- Motion in free fall d- Speed h- Acceleration

2. Taking into account the next table that describes the motion in a straight line of an object: A B C D E F G Time (s) 0 1 2 3 4 5 6 Position (m) 20 25 30 35 40 35 30 a) Draw the position vs time graph. b) Calculate the travelled distance and displacement from A to G c) Average speed and average velocity from A to G. What is the difference between them? s = 30 m; x=10 m; v=5 m/s; v=1,67 m/s 3. An object travels 50 m to the right in 12 seconds. Then it stops for 5 seconds and it needs 20 seconds to come back. Draw the x-t graph and calculate the average speed and velocity. v=2,7 m/s; v=0 m/s 4. A cyclist travels at a constant velocity of 40 km/h on a straight road for half an hour . Then he stops for a quarter of an hour since he is tired and he needs to rest and take a soft drink. Afterwards he starts his way back at a constant velocity, and it takes him 2 hours to reach his starting point. Calculate: a) His velocity on his way back b) Draw s-t and v-t graphs c) Work out the cyclist’s motion equations. v = -2,77 m/s (-10 km/h)

5. Observe the following graph for a motion: x/m) a) What is the speed of the object? b) After 25 seconds where will the object be? 40

c) Draw the graph v-t for this motion. 30

20 v = 5 m/s; x=135 m 10

0 2 4 6 8 10 t(s) 6. A car starts moving at a constant velocity of 50 km/h. Three hours later another car starts moving from the same point towards the first car, at 80 km/h. When and where will the second car reach the first one? Draw the s-t graph of both cars. t = 8 h; x = 400 km 7. A and B stations are 45 km apart. At six o’clock a train leaves form station A at 60 km/h towards B. Half an hour later, another train leaves station B (towards A) at a constant velocity of 90 km/h. When and where will they meet? t = 0,6 h; x = 36 km

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8. The table gives some data for a Ferrari racing car at the start of a Grand Prix race. Plot speed- time graph for the car and calculate its acceleration at: 1s, 6s & 13 s Speed (m/s) 0 10 20 29 37 50 59 64 65 65 Time(s) 0 1 2 3 4 6 8 10 12 14 What kind of motion is it? 10 m/s2; 6,5 m/s2; 0 m/s2 9. The driver of a lorry travelling at 72 km/h sees a huge stone on the road 20 m far from him and pushes the brake. The lorry stops due to an acceleration of 6 m/s2. How long did it take him to stop? Did he crash into the stone? t = 3,33 s 10. An aircraft resting at the airport’s runway needs to travel 600 m for 30 s before taking off. Calculate its velocity at the moment of leaving the runway. v = 40 m/s 11. An aircraft lands at 288 km/h and it needs 2 km before stopping. Calculate the aircraft’s acceleration and the time taken. a = -1,6 m/s2 ; t = 50 s 12. A stone falls from a height of 40 m to the ground. Calculate the time needed to reach the ground and the velocity at that moment. t = 2,82 s ; v = - 27,6 m/s 13. A sack is released at a height of 20 m from a hot-air balloon travelling upwards at a constant rate of 2 m/s. How long will it take the sack to reach the ground? What will it be its velocity at that very moment? v = -20 m/s ; t = 2,2 s 14. Look at these v/t graphs, and work out which one represents the motion of a stone that has been thrown upwards and afterwards falls downwards. v v v

t t t

15. Patrick and Sarah are enjoying themselves on a merry-go-round or carousel. Patrick is sitting on a horse at the periphery, 4 m from the centre. Sarah is in a car 2 m from the centre. The journey takes 3,5 minutes and it completes 9 cycles. Calculate: d) Each angular and linear speed e) Centripetal acceleration f) Period and frecuency ωP = 0,27 rad/s = ωS ; vP = 1,07 m/s; vS = 0,54 m/s 2 2 -2 aP = 0,28 m/s ; aS = 0,14 m/s ; T=23,33 s; f= 4,28.10 Hz

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USM & UASM GRAPHS

I. Motion’s sensor

Aim: To get a motion-graph already drawn by Pasco motion’s sensors.

We will draw an s-t graph with the DataStudio program, and we’ll try to get it moving in front of a sensor (forwards, backwards, stop, speed up….)

Steps to follow: New activity Draw a graph (pencil and curve) Press Start ,and start moving. Press Stop

II. Study of the motion down a sloping surface

Aim: calculate the acceleration of a ball released at the top of an sloping surface.

Assemble this structure (First photoelectric barrier on the top, at the point where the ball will be released)

We will measure the time needed by the ball to travel the whole surface. The results will be collected in the table below. We will carry on the experiment three times in order to reduce the error (random error) made. The value used to calculate the acceleration will be the average of the times collected.

t (s)

tm (s)

Initial position: xo= ………………. a- Calculate the ball’s acceleration and final velocity b- What kind of motion is it? c- What is the initial velocity? d- Draw the s-t & v-t graphs.

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III- Free fall

Assemble the structure shown in the

Experiment 1 1- Measure the initial position. Release the ball and measure the time to reach the table. Repeat the experiment three times.

2- Try it again using balls of different .

1 2

t1(s)

tm (s)

3- Calculate the final velocity for both balls. Compare the results...Are they logical? Why? What kind of motion is it?

Experiment 2 Release the ball from different positions, and measure the time needed.

Initial position (yo) Distance travelled (y- t1 (s) t2 (s) t3(s) taverage (s) t2 (s2) yo)

Draw a s-t2 graph. The slope of the graph is g/2, so we can work out the value of g. Compare it with the real one. Each student will have to write one lab report. Ten-day period to hand in a- Explain the procedure: what have you done? b- Results, graphs... c- Conclusions (answer to the questions)

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Vocabulary  Slope: malda

Informazio iturriak https://www.khanacademy.org/science/ap-physics-1

Angular Speed of Earth Our Earth takes about 365.25 days to finish one revolution around the Sun, now translate days into seconds, T = 365.25 x 24 x 60 x 60 = 31557600 seconds Angular speed = 2π/T Therefore, Hence, ω = 1.99 x 10−7 radians /seconds.

Informazio – iturriak: Khan academy

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