* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download VU2 Movement 2008
Laplace–Runge–Lenz vector wikipedia , lookup
Center of mass wikipedia , lookup
Coriolis force wikipedia , lookup
Faster-than-light wikipedia , lookup
Velocity-addition formula wikipedia , lookup
Hooke's law wikipedia , lookup
Jerk (physics) wikipedia , lookup
Modified Newtonian dynamics wikipedia , lookup
Specific impulse wikipedia , lookup
Seismometer wikipedia , lookup
Electromagnetism wikipedia , lookup
Work (thermodynamics) wikipedia , lookup
Hunting oscillation wikipedia , lookup
Fictitious force wikipedia , lookup
Newton's theorem of revolving orbits wikipedia , lookup
Centrifugal force wikipedia , lookup
Relativistic mechanics wikipedia , lookup
Classical mechanics wikipedia , lookup
Mass versus weight wikipedia , lookup
Equations of motion wikipedia , lookup
Rigid body dynamics wikipedia , lookup
Centripetal force wikipedia , lookup
VCE Physics Unit 2 Topic 1 Movement View physics as a system of thinking about the world rather than information that can be dumped into your brain without integrating it into your own belief systems. Unit Outline To achieve this outcome the student should use scientific methods, data, theories and knowledge to: • • • • • • • • • Describe non-uniform and uniform motion along a straight line graphically; Analyse motion along a straight line graphically, numerically and algebraically; Describe how changes in movement are caused by the actions of forces; Model forces as external actions through the centre of mass point of each body; Explain movement in terms of the Newtonian model and some of its assumptions, including Newton’s 3 laws of motion, forces act on point particles, and the ideal, frictionless world. Compare the accounts of the action of forces by Aristotle, Galileo and Newton. Apply the vector model of forces including vector addition, vector subtraction and components to readily observable forces including weight, friction and reaction forces; Model mathematically work as force multiplied by distance for a constant force and as area under the force versus distance graph. Interpret energy transfers and transformations using an energy conservation model applied to ideas of work, energy and power, including transfers between – kinetic energy and gravitational potential energy close to the Earth’s surface; – potential energy and kinetic energy in springs; Chapter 1 Introduction 1.0 An Ideal World To make life easier for Physics students situations or events which require mathematical analysis are often described as occuring in an ideal, frictionless world. In the ideal world an object under the influence of Earth’s gravity will accelerate at 9.8 ms-2 throughout its journey never reaching a terminal velocity. In the ideal world the laws of motion apply exactly, eg. objects which are moving will continue to move with the same speed unless or until something occurs to change this. In the ideal world energy transformations are always 100% efficient, so that the potential energy of a pendulum at the top of its swing is all converted to Kinetic Energy (motion energy) at the bottom. In the ideal world perpetual motion machines are commonplace. 1.1 The S.I. System In 1960, the “General Conference of Weights and Measures” , a Paris based international organisation, agreed that one set of units would be adopted world wide for the measurement of physical quantities. This system is called the Systeme Internationale d’Units, or more simply the S.I. System. The system is used and recognised worldwide and defines 7 fundamental units. Physical Quantity S.I. Unit Symbol Length metre m Mass kilogram kg Time second s Electric Current ampere A Temperature kelvin K Luminous Intensity candela cd Amount of Substance mole mol All other units are derived from these 7 fundamentals. A derived unit is the force unit, the Newton, which is found from mass x length x 1/(time)2 Thus the Newton has dimensions kg x m x s-2 1.2 Length; metre [m] S.I. Definitions Current; ampere [A] It is the distance light It is that current which produces travels, in a vacuum, a force of 2 x 10-7 N between two in 1/299,792,458th of parallel wires which are 1 metre a second. apart in a vacuum. Temperature; kelvin [K] It is 1/273.16th of the thermodynamic temperature of the triple point of water. Amount of Substance; mole [mol] Mass; kilogram [kg] It is the mass of It is the amount of substance that a platinum-iridium cylinder kept at contains as many elementary Sevres in France. It is now the only units as there are atoms in 0.012 basic unit still defined in terms of a kg of 12C material object. Luminous Intensity; candela [cd] It is the intensity of a source Time; second [s] of light of a specified It is the length of time frequency, which gives a taken for 9,192,631,770 specified amount of power in periods of vibration of a given direction. the caesium-133 atom Questions to occur. Fundamentals QUESTIONS 1. Which of the following quantities have fundamental units and which have derived ? Quantity (Unit) Fundamental Power (Watts) Derived √ Distance (metre) √ Time (second) √ Force (Newton) √ Energy (Joule) √ Mass (kilogram) √ √ Electrical Resistance (ohms) Temperature (kelvin) √ Electric Current (amperes) √ Fundamentals 2. From which of the fundamental units do the following derive their units ? Quantity (Unit) e.g Force (Newton) Acceleration (ms-2) Momentum (kgms-1) Impulse (Newton.second) Velocity (ms-1) Work (Joule) Note: W = F.d Fundamental Units Mass (kg), length (m), time (s) Length (m), time (s) Mass (kg), length (m), time (s) Mass (kg), length (m), time (s) Length (m), time (s) Mass (kg), length (m), time (s) Fundamentals 3. Show that 1 ms-1 = 3.6 kmh-1 Two relevant conversion factors are: 1 km = 1000 m, 1 h = 3600 s These can be written as 1km or 1000m 1000m and 1km 1h or 3600 s 3600s 1h Which ones to use ? Easy, you want to end up with km on the top line and h on the bottom 1m x 1km x s 1000m so 1 ms-1 = 3.6 kmh-1 3.6 3600s 1h 1.3 Position In order to specify the position of an object we first need to define an ORIGIN or starting point from which measurements can be taken. For example, on the number line, the point 0 is taken as the origin and all measurements are related to that point. -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 Numbers to the right of zero are labelled positive Numbers to the left of zero are labelled negative A number 40 is 40 units to the right of 0 A number -25 is 25 units to the left of 0 Questions Position 4. What needs to be defined before the position of any object can be specified ? A zero point needs to be defined before the position of an object can be defined 5. (a) What distance has been covered when an object moves from position +150 m to position + 275 m ? Change in position = final position – initial position = +275 – (+150) = + 125 m. Just writing 125 m is OK (b) What distance has been covered when an object moves from position + 10 m to position -133.5 m ? Change in position = final position – initial position = -133.5 – (+10) = - 143.5 m. Negative sign IS required Chapter 2 Vectors & Scalars 2.0 Scalars and Vectors Before proceeding further we need to define two new quantities: SCALAR QUANTITIES These are completely defined by •A Number and •A Unit Examples of scalars are: Temperature 170, Mass 1.5 kg VECTOR QUANTITIES These are completely defined by •A Number •A Unit and •A Direction Examples of vectors are: Displacement 25 km West, Force 14 Newtons South Vectors are usually represented by an ARROW, with the length of the arrow indicating the size of the quantity and the direction of the arrow the direction of the quantity. N This vector represents a Force of 4 N, acting North West Questions Vectors and Scalars 6. Which of the following quantities are scalars and which vectors ? Quantity Unit Scalar Vector Distance metre Momentum kgms-1 East Kinetic Energy joule Acceleration ms-2 N45oE √ Gravitational Field Strength Nkg-1 downwards √ Displacement metre sideways √ Age years Velocity ms-1 West Temperature oC √ √ √ √ √ √ 2.1 Vector Addition & Subtraction Vectors can be at any angle to one another and still be added. This can be done in two ways: Draw accurate, scale vectors on graph paper and measure the size and direction of the result of the addition, called the “resultant vector” Draw sketch vectors and use trig and algebraic methods to calculate the size and direction of the resultant. The tail of the second adds to the head of the first ADDITION = + 5.0 units SE 5.0 units NE 7.1 units East The resultant is drawn from the tail of the first to the head of the second SUBTRACTION 5.0 units NE _ = 5.0 units SE To subtract, reverse the direction of the negative vector then add. 5.0 units NE + 5.0 units NW = 7.1 units North 2.2 Vector Components A single vector can be broken up into two or more parts called COMPONENTS. This process is useful when, for example, trying to find the vertical and horizontal parts of a force which is accelerating a mass through the Earth’s atmosphere. FH and FV are the COMPONENTS of the force F. F = 5 x 106 N At present, the total force is directed at 30o above the horizontal FV 300 30o FH The Horizontal component of the force (FH) can be found using trig methods: FH = F cos 30o = (5 x 106) ( 0.866) = 4.3 x 106 N Similarly for the Vertical component (FV), FV = F sin 30o = (5 x106)(0.5) = 2.5 x 106 N Questions Vector Addition 7. What is the resultant force when 2 forces (6.0 N west and 4.0 N south) act on an object at the same time ? 6 N west θ θ = tan-1 4/6 = 33.7o 4 N south Resultant Force = √(6)2 + (4)2 = 7.2 N Vector Subtraction 8. Calculate the change in velocity of an object initially travelling at 8.5 ms-1 East whose final velocity was 8.5 ms-1 West. (remember Change in Velocity = Final Velocity – Initial Velocity) 8.5 ms-1 West 8.5 ms-1 West + 8.5 ms-1 East 8.5 ms-1 West 17 ms-1 West = = Vector Components 9. An boy fires a stone from slingshot. The stone leaves with a velocity of 27 ms-1 at an angle 320 above the horizontal. Calculate the vertical and horizontal components of the stone’s velocity. Vv θ = 32o VH VH = 27 Cos 32o = 22.9 ms-1 Vv = 27 sin 32o = 14.3 ms-1 Vector addition/subtraction 10. Calculate the acceleration of a car whose velocity changes from 16 ms-1 west to 21 ms-1 north in 1.5 seconds (acceleration = change in velocity/change in time) Acceleration is a vector quantity so a vector calculation is required to calculate it. Initial Velocity 16 ms-1 West Final Velocity 21 ms-1 North Resultant Velocity = √(21)2 + (16)2 = 26.4 ms-1 Change in Velocity = VF – Vi - = + = θ Acceleration = change in velocity/change in time = 26.4/1.5 = 17.6 ms-2 at N37.3oW θ = tan-1 16/21 = 37.3o Chapter 3 Kinematics 3.0 Distance & Displacement Distance is a SCALAR quantity. It has a Unit (metres) but no Direction. Distance is best defined as “How far you have travelled in your journey” Displacement is a VECTOR quantity Having both a Unit (metres) and a Direction. Displacement is best defined as “How far from your starting point you are at the end of your journey” The difference between these two quantities is easily illustrated with a simple example. You are sent on a message from home to tell the butcher his meat is off. Positive Direction 2 km At this point in the journey , Distance travelled = 2 km and Displacement = + 2 km At the end of the journey, Distance travelled = 2 + 2 = 4 km while Displacement = +2 + (-2) = 0 km 3.1 Speed & Velocity These two terms are used interchangeably in the community but strictly speaking they are different: Velocity is the time rate of change of displacement, i.e., Velocity = Displacement Time Speed is the time rate of change of distance, i.e., Speed = Distance Time Speed is a SCALAR QUANTITY, having a unit (ms-1), but no direction. Thus a speed would be: 100 kmh-1 or, 27 ms-1 Velocity is a VECTOR QUANTITY, having a unit (ms-1) AND a direction. Thus a velocity would be: 100 kmh-1 South or - 27 ms-1 3.2 Acceleration Acceleration is defined as the time rate of change of velocity, i.e., Acceleration = Velocity Time There is no scalar measurement of acceleration, so acceleration MUST always be quoted with a direction. Acceleration is a VECTOR QUANTITY having both a unit (ms-2) and a direction. Typically, Acceleration means an increase in velocity over time, while Deceleration means a decrease in velocity over time. v a When v and a are in the same direction, the car accelerates and its velocity will increase over time. a v When v and a are in the opposite direction, the car decelerates and its velocity will decrease over time. 3.3 Instantaneous & Average Velocity The term velocity can be misleading, depending upon whether you are concerned with an Instantaneous or an Average value. The best way to illustrate the difference between the two is with You take a car journey out of a city to an example. your gran’s place in a country town 90 km away. The journey takes you a total of 2 hours. The average velocity for this journey, vAV = Total Displacement = 90 = 45 kmh-1 Total Time 2 However, your instantaneous velocity measured at a particular time during the journey would have varied between 0 kmh-1 when stopped at traffic lights, to, say 120 kmh-1 when speeding along the freeway. Average and Instantaneous velocities are rarely the same. Unless otherwise stated, all the problems you do in this section of the course require you to use Instantaneous Velocities. Questions Kinematics 11. A runner completes a 400 m race (once around the track) in 21 seconds what is: (a) her distance travelled (in km), (b) her displacement (in km), (c) her speed (in ms-1) and (d) her velocity (in ms-1) ? (a) Distance = 0.4 km (b) Displacement = 0 km (c) Speed = distance/time = 400/21 = 19ms-1 (d) Velocity = displacement/time = 0/21 = 0 ms-1 Acceleration 12. A roller coaster, at the end of its journey, changes it’s velocity from 36 ms-1 to 0 ms-1 in 2.5 sec. Calculate the roller coaster’s acceleration. a = change in velocity/change in time = (0 – 36)/2.5 = - 14.4 ms-2 Chapter 4 Motion by Graphs 4.0 Graphical Relationships It is often useful and convenient to represent information about things like position, velocity, acceleration etc., using graphs. Graphs “tell you a story”. You need to develop the skills andThere are two basic types of graphs used abilities to “read the story”. in Physics: (a) Sketch Graphs – give a “broad brush” picture of the general relationship between the two quantities graphed. (b) Numerical Graphs – give the exact mathematical relationship between the two quantities graphed and may be used to calculate or deduce numerical values. 4.1 Sketch Graphs Sketch graphs have labelled axes but no numerical values, they give a general broad brush relation between the quantities. Velocity Distance Displacement Time The Story: The Story: The object begins its journey The The Story: Story: As time passes itsthe at the origin at t = 0. the As time As As time time passes passes, passes its displacement displacement gets velocity distance remains of the increases at a constant rate larger atfrom an increasing constant. object itsSo time (slope is constant). rate. rate ofis change of of This starting a graph point does an displacement which equals This the graphatof an object not is change. travelling velocity is constant.with object moving constant This the velocity graph of a This is ais graph of an object constant acceleration travelling at constant stationary object velocity Questions Sketch Graphs Distance 13 (a) Distance versus time graph. As time passes displacement remains the same. This is the graph of a stationary object Time Displacement (b) Displacement versus time graph. As time passes its displacement is increasing in a uniform manner. This is a graph of an object travelling at constant velocity. Time Sketch Graphs (c) Velocity Velocity versus time graph. As time passes the velocity of the object remains the same. This is a graph of an object travelling at constant velocity. Time (d) Displacement Time Displacement versus time graph. As time passes its displacement gets larger at an increasing rate. This is a graph of an accelerating object. (if the shape is parabolic the object is increasing its speed in a uniform fashion i.e. it has a constant acceleration) 4.2 Exact Graphical Relationships The graphs you are required to interpret mathematically are those where distance or displacement, speed or velocity or acceleration are plotted against time. The information available from these graphs are summarised in the table given below. Graph Type Distance or Displacement versus Time Speed or Velocity versus Time Acceleration versus Time Obtain from Slope of the Graph Obtain from Area Under the Graph Distance or Displacement Speed or Velocity No Useful Information Speed or Velocity Acceleration Distance or Displacement Acceleration No Useful Information Velocity Read directly from the Graph Learn this table off by heart. Put it on any cheat sheet you are allowed to use. Questions Graphical Interpretation 14. Given below is the Distance vs Time graph for a cyclist riding along a straight path. (a) In which section (A,B,C or D) is Distance the cyclist stationary ? A B C D (b) In which section is the cyclist travelling at her slowest (but not zero) 20 speed ? (c) What is her speed in part (b) above ? 10 (d) What distance did she cover in the first 40 seconds of her journey ? (e) In which section(s) of the graph is Time (s) her speed the greatest ? 0 (f) What is her displacement from 20 30 40 50 60 10 her starting point at t = 50 sec ? (a) Stationary in section C (b) Section B (c) Travels 10 m in 20 s speed = 10/20 = 0.5 ms-1 (d) 20 m (read directly from graph) (e) Section D (travels 20 m in 10 s) speed = 2 ms-1 (f) Displacement at t = 50 s is 0 m (i.e., back at starting point) Graphical Interpretation 15. Shown below is the Velocity vs Time graph for a motorist travelling along a straight section of road. (a) What is the motorist's Velocity displacement after 4.0 sec ? (b) What is the motorists acceleration during this 4.0 sec period ? (c) What distance has the Time(s) motorist covered in the 20.0 sec of his journey ? (d) What is the motorist's (a) Displacement = area under velocity time graph. displacement at t = 20.0 sec Between t = 0 and t = 4 s. Area = ½ (10 x 4) = 20 (e) What happens to the motorists velocity at t = 20.0 m sec? Is this realistic ? (b) Acceleration = slope of velocity time graph (f) Sketch an acceleration vs = (10 – 0)/(4 – 0) = 2.5 ms-2 time graph for this journey. (c) Distance = area under graph (disregarding (ms-1) 10 8 6 4 2 0 1 -2 -4 -6 -8 -10 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 signs) Total area = ½(10 x 4) + (6 x 10) + ½(10 x 2) + ½(9 x 2) + (6 x 9) = 20 + 60 + 10 + 9 + 54 = 153 m (d) Displacement = area under graph (taking signs into account) = ½(10 x 4) + (6 x 10) + ½(10 x 2) ½(9 x 2) - (6 x 9) = 20 + 60 + 10 - 9 – 54 = 27 m (e) Velocity falls from 9 ms-1 to zero in no time – no realistic, as it would require an infinite deceleration to achieve this. Graphical Interpretation 15, continued a (e) To Infinity 2.5 4 -4.5 -5 10 12 14 t 20 Graphical Interpretation 16. An object is fired vertically upward on a DISTANT PLANET. Shown below is the Velocity vs Time graph for the object. The time commences the instant the object leaves the launcher (a) What is the acceleration of Velocity (ms-1) the object ? 30 (b) What is the maximum height attained by the object ? (c) How long does the object take to stop ? Time (s) (d) How far above the ground is 0 2 4 6 8 10 12 the object at time t = 10.0 sec ? -30 (a) Acceleration = slope of velocity time graph. Slope = (30 – 0)/(0 – 6) = -5.0 ms-2 (b) Displacement = area under velocity time graph = ½ (6 x 30) = 90 m (c) Stops at t = 6.0 sec (d) The rocket has risen to a height of 90 m in 6 sec. It then falls a distance of ½ (4 x 20) = 40 m, so it will be 90 – 40 = 50 m above the ground at t = 10 s Chapter 5 The Equations of Motion 5.0 The Equations of Motion The Equations of Motion are a set of equations linking displacement, velocity, acceleration and time. They allow calculations of these quantities without the need for graphical representations. The 3 main equations are: v = u + at Where, v2 = u2 + 2as u = initial velocity (ms-1) s = ut + ½at2 v = final velocity (ms-1) a = acceleration (ms-2) s = displacement (m) t = time (s) THESE EQUATION CAN ONLY BE USED IF THE ACCELERATION IS CONSTANT When using the equations, always list out the information given and note what you need to find, then choose the most appropriate equation. +ve u= v= a= s= t= In some cases you also need to define a positive direction, up or down for vertical motion, left or right for horizontal motion questions 5.1 Motion Under Gravity Objects (close to the surface) The acceleration in this case is falling through the Earth’s ALWAYS directed downward. gravitational field are subject Objects thrown or fired directly to a constant acceleration of upwards would thus have their -2 9.8 ms . velocity and acceleration in Since the acceleration is opposite directions. constant this motion can be analysed by the equations of motion. The calculations using the equations of v = u + at v2 = u2 + 2as s = ut + ½at2 motion always ignore the effects of friction and air resistance You need to go through the same process of listing information and deciding on a positive direction +ve u= v= a= s= t= Questions Equations of Motion 17. A truck travels from rest for 10.0 sec with an acceleration of 3.0 ms-2. Calculate the truck's final velocity and total distance travelled. List information: u = 0, v = ?, a = 3.0 ms-2, s = ?, t = 10 s firstly find v, use v = u + at v = 0 + (3.0)(10) = 30 ms-1 then find x use x = ut + ½at2 (0)(10) +½(3.0)(10)2 = 150 m. 18. A ball rolling down an inclined plane from rest travels a distance of 20.0 m in 4.00 sec. Calculate its acceleration and its final speed List information: u = 0, v = ?, a = ?, x = 20.0 m, t = 4.0 s Firstly find a, use x = ut + ½at2 20.0 = (0)(4.0) + ½a(4.0)2 a = 1.25 ms-2 The find v, use v = u + at v = 0 + (1.25)(4.0) 5.0 ms-1 Equations of Motion 19. The speed of a freewheeling skateboard travelling on a level surface falls from 10.0 ms-1 to 5.00 ms-1 in moving a distance of 30.0 m. If the rate of slowdown is constant, how much further will the skateboard travel before coming to rest ? List information u = 10 ms-1, v = 5.0 ms-1 ,a = ?, x = 30 m, t = ? Cannot get to answer in 1 step. First find acceleration Use v2 = u2 + 2ax a = (v2 – u2)/2x a = - 1.25 ms-2 Now new information u = 5.0 ms-1, v = 0, a = -1.25ms-2, x = ?, t = ? Use v2 = u2 + 2ax x = (v2 – u2)/2a x = 10 m 20. A bullet leaves the barrel of a gun aimed vertically upwards at 140 ms-1. How long will it take to reach its maximum height ? (Ignore air resistance and use g = 10 ms-2) . List information (up is +ve) u = 140 ms-1 v = 0, a = -10 ms-2, x = ? t = ? Use v = u + at t = (v – u)/a = (0 – 140)/-10 = 14 s Chapter 6 Forced Change 6.0 What is a Force ? "A force is an interaction between two material objects involving a push or a pull." How is this different from the usual textbook definition of a Force Forces are like conversations in simply being a “push or a pull” ? that: First, a force is an "interaction". To have a force, you have to You can compare a force to another have 2 objects - one object common interaction - a conversation. pushes, the other gets pushed. A conversation is an interaction between 2 In the definition, "(material) people involving the exchange of words objects" means that both objects (and ideas). have to be made out of matter Some things to notice about a atoms and molecules. They both conversation (or any interaction) are: have to be "things", in the sense To have a conversation, you need two that a chair is a "thing". people. One person can't have a A force is something that conversation happens between 2 objects. It is A conversation is something that happens not an independently existing between two people. "thing" (object) in the sense that It is not an independently existing "thing" a chair is an independently (object), in the sense that a chair is an existing "thing". Questions independently existing "thing". Force 21. A force is an interaction between 2 objects. Therefore a force can be likened to A: Loving chocolate B: Fear of flying C: Hatred of cigarettes D: Having an argument with your partner 22. Between which pair can a force NOT exist ? A: A book and a table B: A person and a ghost C: A bicycle and a footpath D: A bug and a windscreen 6.1 What Kinds of Forces Exist ? For simplicity sake, all forces (interactions) between objects can be placed into two broad categories: 1. Contact forces are types of forces in which the two interacting objects are physically contacting each other. Examples of contact forces Force is a quantity which is include frictional forces, measured using the derived tensional forces, normal metric unit known as the forces, air resistance forces, Newton. and applied forces. One Newton (N) is the amount of force required to give a 1 kg 2. Field Forces are forces in which mass an acceleration of 1 ms-2. the two interacting objects are not So 1N = 1 kgms-2 in contact with each other, yet are able to exert a push or pull despite Force is a vector quantity, a physical separation. you must describe both the Examples of field forces include magnitude (size) and the Gravitational Forces, Electrostatic direction. Forces and Magnetic Forces Questions Contact or Field Forces 23. Classify the following as examples of either Contact or Field forces in action (or maybe both acting at the same time). EXAMPLE CONTACT FORCE (a) A punch in the nose √ (b) A parachutist free falling √ (c) Bouncing a ball on the ground √ FIELD FORCE √ (d) A magnet attracting a nail √ (e) Two positive charges repelling each other √ (f) Friction when dragging a refrigerator across the floor √ (g) A shotput after leaving the thrower’s hand √ √ 6.2 What Do Forces Do ? Forces affect motion. They can: BEGINNING MOTION: A constant force (in the same direction as the motion) produces an ever increasing velocity. • Begin motion • Change motion • Stop motion • Have no effect FR NO EFFECT: A total applied force smaller than friction will not move the mass CHANGING MOTION: A constant force (at right angles to the motion) produces an ever changing direction of velocity. STOPPING MOTION: A constant force (in the opposite direction to the motion) produces an ever decreasing velocity. 6.3 Where Forces Act Forces acting on objects must have a point of application, a place where the force acts. For Contact Forces the point of application is simply the point at which the force initiator contacts the object. Force of carton on finger Force of finger on carton C of M For Field Forces, the only one applicable in movement being gravity, will act through the centre of mass of the object Gravitational Force Questions Net Force 24. A body is at rest. Does this necessarily mean that it has no force acting on it ? Justify your answer. NO – A body will remain at rest if the NET FORCE acting is zero – it could have any number of forces acting on it. So long as these forces add to zero it will remain at rest. 25. Calculate the net force acting on the object in each of the situations shown. (a) (b) 900 N 1200 N N 300 N Left (c) 0N 75 N 95 N 20 N Left 250 N 250 N (d) 300 N Down 150 N 450 N 6.4 Forces in 2 Dimensions Forces can act in any direction and the total or resultant force is the vector sum of all the forces acting. Tim, Tom and Tam, the triplets, are fighting over a teddy bear. Each exerts a different force. What will be the net force on the bear ? Tom Tim Tam A force diagram shows each boy’s contribution Add the vectors head to tail. FTOM = 25 N FTIM = 26 N Tim FTAM = 18 N FRESULTANT The resultant force is the vector joining the starting point to the finishing point The bear will then accelerate in the direction of the resultant force Questions Net Force 26. Tim, Tom and Tam, the triplets, are fighting over a teddy bear. Each exerts a different force. A force diagram shows each boy’s contribution. What will be the net force on the bear ? FTIM = 35 N Resolve FTIM and FTOM to give F = 9 N left Then resolve this force with FTAM 9N θ X 12 N FTOM = 26 N FTAM = 12 N X = √(92 + 122) = 15 N θ = sin-1 (9/12) = 48.60 Net force is 15 N directed at S48.6oW 6.5 Weight Weight is the outcome of a gravitational field acting on a mass Weight is a FORCE and is measured in Newtons. Its direction is along the line joining the centres of the two bodies which, between them, generate the Gravitational Field. 1 kg 9.8 N Near the surface of the Earth, each kilogram of mass is attracted toward the centre of the earth by a force of 9.8 N. (Of course each kilogram of Earth is also attracted to the mass by the same force, Newton 3) So, the Gravitational Field Strength near the Earth’s surface = 9.8 Nkg-1 Weight and mass are NOT the same, but they are related through the formula: W = mg Where: W = Weight (N) m = mass (kg) g = Grav. Field Strength (Nkg-1) Questions 1 kg Mass & Weight 27. Fill in the blank spaces in the table based on a person whose mass on earth is 56 kg Planet Mass on planet (kg) Grav Field Strength (Nkg-1) Earth 56 9.81 Mercury 56 0.36 20.2 Venus 56 0.88 20.2 Jupiter 56 26.04 1458.2 Saturn 56 11.19 626.6 Uranus 56 10.49 Weight on planet (N) 549.4 587.4 6.6 Reaction Force All objects on and near the Earth’s surface are subject to the gravitational force. Any object subject to a net or resultant force will accelerate in the direction of that force (Newton 2). Why then do objects placed on a table on the Earth’s surface remain stationary ? There must be a force equal in size and opposite in direction to cancel out the gravitational force. There is such a force. It is called the REACTION or NORMAL FORCE. R W W Because there is no net or resultant force on the vase, it remains stationary on the table Remove the table, the reaction force disappears and the vase accelerates under the action of W, until it encounters the floor and probably smashes. The Reaction Force only exists as a result of the action of the weight of the vase acting on the table top and as such the reaction force does not exist as an isolated force in its own right. Note: W and R are NOT an action reaction pair. Why? Because when R disappears W does not. Chapter 7 Centre of Mass 7.0 Centre of Mass In order to deal with large objects it is useful to think of all the object’s mass being concentrated at one point, this point being the Centre of Mass of the object. For regularly shaped objects eg. squares or rectangles, cubes or spheres the Centre of Mass of the object is in the C of M geometric centre of the object For odd shaped objects such as a boomerang, the Centre of Mass may Centre of Mass fall outside the perimeter of the object. The C of M is the point around which the object it will spin if a torque or turning force is applied to the object. 7.1 Translation and Rotation When a Force acts C through the Centre of Mass (of M) of an object or structure, it causes Translational Motion, ie. The object moves in the direction of the applied force according to Newton’s 2nd Law. (see slide 6.3) A Force acting through the Centre of Mass causes Translational Motion only When the force is applied to another part of the object or structure, a TORQUE or TWISTING FORCE or TURNING MOMENT is applied and Rotational as well as Translational motion occurs A Force acting at a point other than the C of M will cause BOTH Translational AND Rotational Motion. Chapter 8 Newton’s Laws 8.0 Aristotle to Newton It was Galileo (1564 – 1642) who Attempts to explain the “causes of motion” (a field of study called dynamics), were first recorded in the time of the ancient Greek philosopher Aristotle (384 – 322 BC). Aristotle It was believed that constant speed required a constant force. This seemed logical as everyone could see that a horse needed to apply a constant pull to haul a cart at constant speed. Galileo However there were problems with the theories which could not, for example, explain why falling objects tended to increase their speed in the absence of any visible force or why heavenly bodies behaved differently than those on earth. Newton was the first to define the property of matter we call INERTIA, (matter’s tendency to resist changes in its motion), with his law which said “when no force exists a body will stay at rest or move with constant speed Philosophers prior to Newton believed a set of laws covering motion on earth could be developed, but they needed to be modified to explain the motions of heavenly bodies. Isaac Newton (1642 – 1727) was the first to realise there WAS a universal set of laws which could describe the motion of ALL bodies, BUT these laws had to be modified for use within the friction riddled confines of the Earth and its atmosphere. Questions Aristotle to Newton Scientist Statement Newton Constant speed requires constant force Aristotle Defined the property of matter called inertia Galileo A universal set of laws applicable everywhere but must be modified for use on earth 8.1 Newton’s Laws Newton developed 3 laws which cover all aspects of motion (provided objects travel at speeds are well below the speed of light). Law 1 (The Law of Inertia) A body will remain at rest, or in a state of uniform motion, unless acted upon by a net external force. Law 2 The acceleration of a body is directly proportional to net force applied and inversely proportional to its mass. Mathematically, a = F/m more commonly written as F = ma Law 3 Newton, at age 26 (Action Reaction Law) For every action there is an equal and opposite reaction. Motion at or near the speed of light is explained by Albert Einstein’s Theory of Special Relativity. 8.2 Objects want to keep on doing what they are doing Newton’s Newton’s 1st Law states: A body will remain at rest, or in a state of uniform motion, unless acted upon by a net external force. Another way of saying this is: st 1 Law If NO net external force exists No Net Force means No Acceleration There is no experiment that can be performed in an isolated windowless room which can show whether the room is stationary or moving at constant velocity. Newton 1 deals with non accelerated motion. It does not distinguish between the states of “rest” and “uniform Most importantly: It requires an motion” (constant unbalanced force Force is NOT needed to velocity). to change the keep an object in motion velocity of an As far as the law is object concerned these are the same thing (state). Is this how you understand the world works ? 8.3 Newton’s nd 2 Law Using the Newton’s 2nd Law states: formula The acceleration of an object as FNET = ma is produced by a net force is only valid for directly proportional to the situations magnitude of the net force FNET, where the in the same direction as the net mass remains force, and inversely proportional constant to the mass of the object. Mathematically, a = FNET/m more Newton actually expressed his 2nd law in terms of momentum. commonly written as FNET = ma The Net Force on Newton 2 deals with accelerated an object equals motion. the rate of change of its FNET is the VECTOR SUM momentum of all the forces acting on an object. Momentum (p) = mass x velocity The acceleration and FNET are ALWAYS in the So, FNET = change in momentum = Δp = mΔv = ma same direction. change in time Δt Δt 8.4 Newton’s 3rd Law Newton's 1st and 2nd Laws tell you what forces do. Newton's 3rd Law tells you what forces are. 2. People associate action/reaction with "first an action, then a reaction” For example, first Suzie annoys Johnnie (action) then For every action Johnny says "Mommy! there is an equal and opposite Suzie’s annoying me!" reaction (reaction). This statement is correct, This is NOT an example what but terse and confusing. is going on here! You need to understand The action and reaction that it means: forces exist at the same time. "action...reaction" means that "equal" means : forces always occur in pairs. Both forces are equal in magnitude. Single, isolated forces never Both forces exist at exactly the same time. happen. They both start at exactly the same instant, "action " and "reaction " and they both stop at exactly the same are unfortunate names for a instant. couple of reasons : They are equal in time. 1. Either force in an interaction "opposite" means that the two forces can be the "action" force or the always act in opposite directions "reaction" force. exactly 180o apart. Questions Newton’s Laws 29. At what speeds are Newton’s Laws applicable ? At speeds way below the speed of light 30. Newton’s First Law: A: Does not distinguish between accelerated motion and constant velocity motion B: Does not distinguish between stationary objects and those moving with constant acceleration C: Does not distinguish between stationary objects and those moving with constant velocity D: None of the above 31. Newton’s Second Law: A: Implies that for a given force, large masses will accelerate faster than small masses B: Implies that for a given force, larger masses will accelerate slower than smaller masses C: Implies that for a given force, the acceleration produced is independent of mass D: Implies that for a given force, no acceleration is produced irrespective of the mass. Newton’s laws 32. Newton’s Third Law: A: Does not distinguish which force of a pair is the “action” force and which is the “reaction” force. B: Implies that both action and reaction forces begin and end at the same instant C: Implies that forces always exist in pairs D: All of the above. 33. Which of Newton’s Laws require that the vector sum of all the forces acting is needed before a calculation of acceleration can be made ? A: Newton’s 1st Law B: Newton’s 2nd Law C: Newton’s 3rd Law D: Newtons 1st and 2nd Laws Newton’s 2nd Law 34. A car of mass 1250 kg is travelling at a constant speed of 78 kmh-1 (21.7 ms-1). It suffers a constant retarding force (from air resistance, friction etc) of 12,000 N (a) What is the net force on the car when travelling at its constant speed of 78 kmh-1 ? At constant velocity, acc = 0 thus ΣF = 0 (b) What driving force is supplied by the car’s engine when travelling at 78 kmh-1 ? At constant velocity ΣF = 0, so driving force = retarding force = 12,000 N (c) If the car took 14.6 sec to reach 78 kmh-1 from rest , what was its acceleration (assumed constant) ? Use eqns of motion u = 0 ms-1 , v = 21.7 ms-1, a = ?, x = ?, t = 14.6 s use v = u + at -> 21.7 = 0 + 14.6(a) -> a = 1.49 ms-2 8.5 The Horse and Cart Problem If the horse and cart exert equal and opposite forces on each other, how come the combination can move ? An explanation hinges on a couple of simple points: (Lets assume no friction) FHC FCH 1. An object accelerates (or not) because of the forces that push or pull on it. (Newton 2) 2. Only the forces that act on an object can cancel. Forces that act on different objects FRH FHR don't cancel - after all, they affect the Why does the cart accelerate? motion of different objects! Looking at the cart alone, just one Why does the horse accelerate? force is exerted on it, (FHC) - the There are 2 forces acting on the horse. force that the horse exerts on it. The cart pulls the horse backwards (FCH), The cart accelerates because the and the road pushes the horse forward (FRH). horse pulls on it! The net force is the vector sum of these two The cart’s acceleration equals the forces. net force on it divided by its mass The horse’s acceleration equals the net force There are 2 pairs of Newton 3 on it divided by the its mass. forces in this situation: If F NET on the horse is zero, what happens ? FHC and FCH The obvious answer is the horse and cart are at rest. FHR and FRH BUT, they could also be moving at constant speed ! Questions Newton 1 Newton’s 3rd Law 35. Explain why, if a cart exerts an equal an opposite force on a horse as the horse exerts on the cart, the combination is able to move forward. It is the forces that act on the individual components (i.e., on the horse or cart individually) that decide whether each will move. If the friction of the road on the horse’s feet is larger than the force of the cart on the horse then the horse will accelerate. If the force of the horse on the cart is greater than the frictional force acting on the cart then the cart will accelerate. 36. A car mass 1500 kg is towing a trailer of mass 750 kg. The car/trailer combination accelerate at 3.4 ms-2. The trailer suffers a constant retarding force of 500 N, while the car suffers a constant retarding force of 1000 N. Calculate the net force acting on the trailer. The net force is that force that provides the acceleration. From Newton 2, ΣF = ma ΣF = (750)(3.4) = 2550 N (2.55 x 103 N) Calculate the driving force supplied by the car’s engine. The driving force must (i) overcome friction and (ii) provide extra force to accelerate the combination ΣF = (1000 + 500) + ma = 1500 + (1500 + 750)(3.4) = 1500 + 7650 = 9150 N (9.15 x 103 N) 8.6 Momentum and Impulse Newton described Momentum as the “quality of motion”, a measure of the ease or difficulty of changing the motion of an object. Momentum is a vector quantity having both magnitude and direction. Mathematically, p = mv Where, p = momentum (kgms-1) m = mass (kg) v = velocity (ms-1) In order to change the momentum of an object a mechanism for that change is required. This mechanism of change is called Impulse. Where, Mathematically, I = Impulse (N.s) I = Ft F = Force (N) t = Time (s) The relationship between momentum and impulse can be derived from Newton’s 2nd Law: F = ma and a = v/t, so F = mv/t Rearranging we get: Ft = mv ie. Impulse = Momentum 8.7 Conservation of Momentum The concept of Momentum is particularly useful in analysing collisions. This is because of the Law of Conservation of Momentum which states: IN AN ISOLATED SYSTEM, TOTAL MOMENTUM IS CONSERVED. The term “isolated system” means no external forces are acting in the situation under investigation. In a crash situation, where the vehicle comes to a halt after, say, hitting a tree, both its velocity and momentum fall to zero. The apparently “lost” momentum, has, in fact, been transferred via the tree to the Earth. Since the Earth has a huge mass (6 x 1024 kg). The change in its velocity is so small as to be negligible. In the crash mentioned, the momentum change is a fixed quantity so the Impulse (the product of F and t) is also a fixed quantity. However the individual values of F and t can vary as long as the multiply to give that fixed value. If t, the time during which the crash occurs, can be lengthened, then the force which needs to be absorbed by the car and its occupants is reduced. Modern vehicles use this concept in crumple zones and air bags as both are designed to extend the time and so reduce the force. Questions Momentum 37. A car (and its occupants) is of total mass of 2250 kg and is travelling at 50 kmh-1 . Approaching, head on, is a motorcycle (and rider) of total mass 350kg travelling at 180 kmh-1 (a) Which vehicle (car or bike) has the greater momentum ? Firstly need to convert speeds to ms-1 50 kmh-1 = 13.9 ms-1 : 180 kmh-1 = 50.0 ms-1 p4wd = mv = (2250)(13.9) = 31275 kgms-1 pcycle = mv = (350)(50) = 17500 kgms-1 So 4WD has the greater momentum. (b) They collide head on and stick together. What velocity will the “wreck” have immediately after collision ? Assume 4WD is travelling to the right and motorcycle to the left. So Σp = p4wd - pcycle = 31275 – 17500 = 13775 kgms-1 to the right. If the vehicles stick together total mass = 2250 + 350 = 2600 kg So Σp = mtotalv v = Σp/mtotal = 13775/2600 = 5.3 ms-1 to the right Momentum and Impulse 38. While talking on a mobile phone a truck driver loses concentration and runs off the road and hits a tree. His speed goes from 20 ms-1 to 0 ms-1 in 0.7 sec. his truck has a mass of 42 tonnes (1 tonne = 1000 kg) (a) Calculate his change in momentum Change in momentum = final mom – initial mom = mvfinal - mvinitial = m (vfinal – vinitial) = 4.2 x 104(0 – 20) = - 8.4 x 105 kgms-1 (negative sign can be omitted from answer as it only indicates direction of momentum change) (b) Calculate the Impulse during the collision Since change in momentum = impulse. Impulse = 8.4 x 105 Ns (c) Calculate the force he will experience during the collision I = Ft F = I/t = (8.4 x 105)/0.7 = 1.2 x 106 N Airbags/Crumple Zones 39. Explain why, in a modern car equipped with seat belts and an air bag , he would likely survive the collision whereas in the past, with no such safety devices, he would most likely have been killed. The change in momentum in any collision is a fixed value thus impulse is also fixed, but the individual values of F and t can vary as long as their product is the that fixed value. In modern vehicles seat belts and crumple zones are designed to increase to time it takes to stop thus necessarily reducing the force needed to be absorbed by the driver because Impulse = Ft. This reduced force will lead to reduced injuries. In the old days the driver would have been “stopped” be some hard object like a metal dashboard and his time to stop would have been much shorter and thus the force experienced would have been larger leading to more severe injury and likely death. Chapter 9 Work, Energy & Power 9.0 Work In Physics, the term WORK is very strictly defined. When a force moves an object through a distance, work has been done. Mathematically: W=Fxd Work is a SCALAR quantity, meaning it has a magnitude but no direction. Where, W = Work (Joules) F = Force (N) d = distance (m) If a force is applied and the object does not move, NO WORK has been done. If the force applied is constant, the work done can be calculated from the formula, W = F x d But, if the force varies during the course of doing the work, as in compressing a spring, the work must be calculated from the area under the force versus distance graph Force Area = Work done Distance Questions Work 40. Calculate the work done on a refrigerator when a net force of 125 N acts over a distance of 4.5 m Work = Fxd = (125) x (4.5) = 562.5 J (5.63 x 102 J) 41. The graph shows the force required to compress a spring Force 6000 (kN) 4500 (a) Calculate the work done in compressing 3000 the spring by 3.0 cm. 1500 Work done = area under graph = ½ base x height Distance (cm) -2 6 1.0 2.0 3.0 4.0 = ½ (3.0 x 10 )(4.5 x 10 ) = 1.35 x 105 J (b) Calculate the further work required to compress the spring from 3.0 cm to 4.0 cm Further work to compress from 3.0 cm to 4.0 cm = area under graph between these two distances = area of trapesium = ½ (height(1) + height(2)) x base = ½ (4.5 x 106 + 6.0 x 106 )(1.0 x 10-2) = 5.25 x 104 J 9.1 Work and Energy It is very easy to say what energy can do, but very difficult to define exactly what energy is. The concept of WORK was developed BY PHYSICISTS as a The relation between work and means of quantifying and energy is summarised by one measuring ENERGY. simple but powerful statement: WORK DONE = ENERGY TRANSFERRED If work has been done on an object, the amount of energy it has MUST have increased. By how much ? By exactly the amount of work done on the object. If an object has done some work, the amount of energy it has MUST have decreased. By how much ? By exactly the amount of work done by the object. Questions Work & Energy 42. How much energy is stored in the spring in question 41 when it has been compressed by 2.0 cm Work Done = ½ base x height = ½ (2.0 x 10-2)(3.0 x 106) = 3.0 x 104 J Since Work Done = Energy Transferred, the energy stored in the spring = 3.0 x 104 J 9.2 Kinetic Energy Kinetic Energy is the energy possessed by moving objects. It is called the “Energy Of Motion”. Kinetic Energy is a SCALER quantity. Mathematically: K.E. = ½mv2 Where: K.E. = Kinetic Energy (Joule) m = mass (kg) v = speed (ms-1) Arrow has K.E. due to its motion Horse has K.E. due to its movement Gears have K.E. due to their rotation 9.3 Gravitational Potential Energy Gravitational Potential Energy, often just called Potential Energy, is the energy possessed by an object due to its position. It is called the “Energy of Position” Where: Potential Energy is a SCALAR quantity. P.E. = Potential Energy (Joules) m = mass (kg) Mathematically: g = Grav. Field Strength (Nkg-1) P.E. = mgh h = height (m) Potential Energy needs a zero point for the measurement of the height, h. The zero point is usually, but not always, the surface of the Earth. The zero point needs to be known for the calculation to have meaning. The man has P.E. due to his height above the ground Skier has P.E. while he is on the slope + KE due to his speed Questions Kinetic & Potential Energy 43. A cyclist is riding her bike along a flat road. She and her bike have a mass of 105 kg. she is travelling at a constant speed of 15 ms-1. (a) Calculate her Kinetic Energy KE = ½ mv2 = ½ (105)(15)2 = 1.18 x 104 J She accidentally rides over a 15 m cliff. (b) What is her potential energy at the top of the cliff ? (take g = 10 ms-2) PE = mgh = (105)(10)(15) = 1.58 x 104 J (c) If all the PE she had at the top of the cliff is converted to KE at the bottom, calculate her vertical speed just before she hits the ground. PETOP = KEBOTTOM 1.58 x 104 = ½ (105) v2 v = 17.3 ms-1 9.4 Hooke’s Law Developed by English scientist Robert Hooke in 1676, the law states that the Restoring Force in an elastic material is directly proportional to its extension. The spring constant (k) is a measure of the nature or quality of the elastic material. The higher its value the greater is the restoring force for a given extension. Mathematically: F = - kx Where: F = Restoring Force (N) k = Spring Constant (Nm-1) x = Extension (m) The negative sign in the equation indicates that the restoring force and the extension are in opposite directions. Questions Hooke’s Law 44. A spring of length 100 cm and spring constant 2.5 x 102 Nm-1 hangs vertically from a retort stand. A total mass of 15.6 kg is hung from the spring. Calculate the extent of the spring’s extension under this load. (Take g = 10 Nkg-1) F = -kx: Force produced by hanging mass = mg = (15.6) (10) = 156 N. So Restoring Force = 156 N x = -F/k = -156/(2.5 x 102) = -0.62 m (could do this calculation without negative sign) by using applied force in Hooke’s Law equation) 45. Calculate the amount of elastic potential energy stored in the extended spring in question 44. EPE = ½ kx2 = ½ (2.5 x 102) (0.62)2 = 48.1 J 9.5 Energy Transfers The Law of Conservation of Energy says: ENERGY CANNOT BE CREATED OR DESTROYED BUT ONLY TRANSFERRED FROM ONE FORM TO ANOTHER. When doing problems concerning energy and energy transfers, it is assumed that the transfers are 100% efficient, meaning no energy losses occur. For instance, a roller coaster will have a large Gravitational Potential Energy component at its highest point most of which will have been converted to Kinetic Energy at its lowest point and in calculations a direct equality can be made between P.E. at the top and K.E at the bottom In real life, these types of conversion processes are never 100% efficient. Friction and the production of heat and sound mean significant losses occur. The efficiency of energy transfer processes can be calculated from: % Eff = Energy Out x 100 Energy In 1 9.6 Elastic Potential Energy Energy stored in springs is called elastic potential energy • • Elastic materials store energy when they are deformed and release that energy when they return to their original condition. The amount of energy stored can be found from the Elastic Potential Energy Formula: ES = ½kx2 where ES = Elastic P. E. (J ) k = Spring Constant (Nm-1) x = extension and or compression (m) “REGULAR” ELASTIC BEHAVIOUR Slope = Spring Force constant (k) F Extension x Area = ½Fx = ½kx2 = Energy stored up to extension x The stored energy can be used to increase the kinetic energy of the “arrows” such as used in cross bows Questions 9.7 Energy Transfers & Heat In our friction riddled world, no energy transfer process is 100% efficient. The end result of the action of these frictional effects is the production of what is called “low grade, unrecoverable heat” This means the heat energy cannot be harnessed to do any useful work, such as boil water or run a pump. Large sources of this kind of heat are the exhausts from fossil fuelled transport such as cars, trucks and trains and from the electricity generation industry. 9.8 Power Power is the time rate of doing work, and since work and energy are equivalent is also the time rate of energy transfer. Mathematically: P = W/t = E/t Since P = W/t and W = F.d, we can say P = F.d/t but d/t = v so P = F.v Where: P = Power (W, Watts) W = Work (J) E = Energy (J) t = time (s) So, the power for a body moving at constant velocity can be found in a one step calculation. Questions Power 46. A train of mass 1.5 x 104 kg is travelling at a constant speed of 70 kmh-1. If the engine is providing a driving force of 3.0 x 107 N , at what power is the engine operating ? 70 kmh-1 = 19.4 ms-1 Power = F.v = (3.0 x 107)(19.4) = 5.82 x 108 W 47. A roller coaster moves through its journey from A to F. The coaster has no motor and is not powered after it leaves point A. Its total mass is 650 kg. The heights above ground of each portion of the track are given below. Positio n A E C B D F Height above Ground (m) A 25.0 B 5.0 C 12.5 D 12.5 E 15.0 F 7.5 Energy Transformations (a) Between which points is (i) the Kinetic Energy increasing, (ii) the Potential Energy falling, (iii) the coaster accelerating (i) AB, EF (ii) BC, DE (iii) All except CD (b) At which point is the force exerted on the coaster by the track at its greatest ? Calculate the Gravitational Potential Energy at Point A. (take g = 10 ms-2) B (At this point the track must supply the reaction force to counteract the weight of the coaster AND the centripetal force to make the coaster go round the ‘corner” (c) Calculate the Gravitational Potential Energy at Point A. (take g = 10 ms-2) PE = mgh = (650)(10)(25) = 1.63 x 105 J (d) Assuming no frictional losses, calculate the Kinetic Energy at point F Loss in PE between A and F = gain in KE between A and F PELOSS = mgh1 – mgh2 = (650)(10)(25 – 7.5) = 1.14 x 105 J (e) Calculate the speed of the coaster at point F At point F, KE = 1.14 x 105 = ½ mv2 v = 18.7 ms-1 Ollie Leitl 2008