DO PHYSICS ONLINE. WEB activity: Use the web to find out more about: Aristotle, Copernicus, Kepler, Galileo and Newton.

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1 DO PHYSICS ONLINE DISPLACEMENT VELOCITY ACCELERATION The objects that make up space are in motion, we moe, soccer balls moe, the Earth moes, electrons moe, Motion implies change. The study of the motion of objects is called kinematics. We will construct simple models starting with the motion of a single object. The object will be modelled as a point particle in space and who dimensions are ignored. A particle is represented in a diagram as a dot. Since the beginning of human time the study of the motions of the heaens has been important. People who had knowledge of the motion of the heaens had power and control oer others because they knew about the seasons, eclipses and could make predictions. In ancient cultures, people were ery frighted during solar and moon eclipses. Deeloping theories about the motion of the Sun and planets was ery frustrating. Many religions had the Earth at the centre of the Unierse. For presenting iews opposing the Earth centred Unierse one could be burnt alie at the stake. Deeloping a consistent theory of motion was difficult. Physical theories are not just discoered, they are creations of the human mind and must be inented and not discoered. It is claimed that the study of physics is man s greatest intellectual achieement. Physics is a creatie process and physicists build on the best of the past to create new isions of the future. A significant time for science was when Nicholas Copernicus ( ) concluded from experimental data on the motion of the planets, that the Earth reoled around the Sun. Johannes Kepler ( ) was able to decipher accurately and precisely the paths of the planets and produced mathematical laws accounting for the motion of the planets around the Sun. The work of Galileo Galilei ( ) produced a dramatic change in the way science was to be done. Galileo s ideas oerturned the Aristotelian iew of the world that had persisted for more than 2000 years. Galileo s genius was that he realized that the world can be described through mathematics and that that experimentation and measurement were crucial elements in deeloping scientific theories where approximation were necessary to limit complexity. Following on from Galileo was Isaac Newton ( ), who deeloped the underlying theory of the causes of motion (dynamics). He published one of the most famous books of science in 1686, Principia. WEB actiity: Use the web to find out more about: Aristotle, Copernicus, Kepler, Galileo and Newton. Group discussion: Why did people such as priests who had scientific knowledge of the seasons and astronomy hae control oer the bulk of the population in their societies? Group discussion: In the period from the 15 th century to the 19 th century in western Europe there was a dramatic change in social, cultural and technological aspects for people but this did not occur in China, which at the beginning of the 15 th century could be consider to be more adanced socially, culturally and technologically than Europe. Explain why the no change in China compared with the scientific and technological deelopments that occurred in Western Europe in this 400 year period.

2 DISTANCE AND DISPLACEMENT In our study of kinematics will we only consider the motion of objects in one-dimension. This is called rectilinear motion. It is a relatie straight forward task to extend the knowledge one gains in studying one-dimensional motion to two- and three-dimensions. One adantage of studying one-dimensional motion is that we don t hae to use ector notation although displacement, elocity and acceleration are ectors. The sign of the physical quantity in this instance gies the direction of the ector. Consider the displacement ectors shown in figure (1). You can t associate a positie or negatie sign with a ector quantity unless it has a direction parallel to one of the coordinate axes. For example, a component of a ector which is parallel to x-axis in a direction pointing towards +x is assigned a positie number and a component pointing towards the x is assigned a negatie number. +y +x s 2 s 1 s 1y s y2 s 1x For the components only: Since the components of the ectors are directed along the coordinate axes we can assign a positie or negatie alue to the components of a ector. The sign then gies the direction. Can not gie the ectors s1 s2 a + or - sign s 2x s s s 1x 1x 1x s s s 1y 1y 1y s s s 2x 2x 2x s s s 2 y 2 y 2 y Fig. 1. axis. A ector can t be a positie or negatie number unless it is parallel to a coordinate Before reading further, iew the animation of the rectilinear motion of a tram that traels to the right then back to the left. Think about the scientific language that you will need to describe the motion of the tram View the animation of a tram To describe the motion of a moing object you must first define a frame of reference (origin and x-axis) and the object becomes a particle. Consider the tram moing backward and forwards along a straight 2.00 km track. The x-axis is taken along the track with the origin at x = 0. The left end of the track is at x =.00 km and the right end of the track is at x = km. The tram start at time t 1 at the left end of the track moes to the right where it stops at the right end and returns to the left end of the track arriing at a later time t 2. At any time t, the position of the tram is gien by its x-coordinate as shown in figure (2).

3 an object becomes a particle represented by a dot -x 0 position x (km) +1 +x Fig. 2. A frame of reference for the rectilinear motion of the tram with the origin at x = 0. The trams traels from the left to the right end and traels back to the left end of the track. At time t1 0 the tram starts its journey at position x km. At time t hours the train completes one cycle to the right end of the track and back again where the final position of the train is x2 1.00km. The time interal t for the journey is t t2t h = 900 s is the Greek letter Delta meaning change in or increment. Note: Time and time interal are different physical quantities. The distance traelled d = 4.00 km. d is The magnitude of the displacement the final position x 2 s x2 x1=.00 (.00) km = 0 km s is the straight line distance between the initial position x 1 and Distance traelled (scalar) is not the same physical quantity as displacement (ector)

4 SPEED AND VELOCITY Speed and elocity refer to how fast something is traelling but are different physical quantities. Also, we need to distinguish between aerage and instantaneous quantities. The aerage speed defined by equation (1) (1) ag ag of a particle during a time interal t in which the distance traelled is d t definition of aerage speed The aerage speed is a positie scalar quantity and not a ector. d is The aerage elocity particle in the time interal (2) ag ag of a particle is defined as the ratio of the change in position ector s of the s t t during which the change occurred as gien by equation (2) definition of aerage elocity For one-dimensional motion directed along the x-axis we do not need the use the ector notation shown by the arrow aboe the symbol. In the example of our tram: aerage speed t t2t h 900 s d = 4.00 km = m. s x2 x1=.00 (.00) km = 0 km = 0 m ag ag aerage elocity ag d 4 km.h t d 410 t 900 m.s s 0 km.h 0 m.s t 16.0 km.h 4.44 m.s S.I. units Note: ery different alues for the aerage speed and aerage elocity. Often the same symbol is used for speed and elocity. You always need to state whether the symbol represents the speed or elocity. Our tram speeds up continually until it reaches the origin, it then slows down and stops at the right end. In the return journey, it continually speeds up until it reaches the origin it then slows down and stops at the left end of the track. The speed and elocity are always changing. Hence, it is more useful to talk around instantaneous alues rather than aerage alues. The instantaneous elocity is a ector quantity whose magnitude is equal to the instantaneous speed and who instantaneous direction is in the same sense to that in which the particle is moing at that instant. The elocity ector is always tangential to the trajectory of the particle at that instant. Remember, speed and elocity are not the same physical quantity een though they hae the same S.I. unit [m.s ].

5 The aerage elocity gien by equation (2) is ag s t If we make the time interal t smaller and smaller, the aerage elocity approaches the instantaneous alue at that instant. Mathematically it is written as limit t 0 s t This limit is one way of defining the deriatie of a function. The instantaneous elocity is the time rate of change of the displacement (3) ds definition of instantaneous elocity dt For rectilinear motion along the x-axis, we don t need the ector notation and we can simply write the instantaneous elocity as (4) dx rectilinear motion dt You don t need to know how to different a function, but you hae to be familiar with the notation for differentiation and be able to interpret the process of differentiation as finding the slope of the tangent to the displacement s time graph at one instant of time figure (3). Fig. 3. The aerage and instantaneous elocities can be found from a displacement s time graph. The slope of the tangent gies the instantaneous elocity (deriatie of a function). The reerse process to differentiation is integration. Graphically, integration is a process of finding the area under a cure. The slope of the tangent to the displacement s time graph is equal to the elocity. The area under a elocity s time graph in a time interal s in that time interal as shown in figure(5). t is equal to the change in displacement When you refer to the speed or elocity it means you are talking about the instantaneous alues. Therefore, on most occasions you can omit the word instantaneous.

6 ACCELERATION Velocity is related to how fast an object is traelling. Acceleration refers to changes in elocity. Since elocity is a ector quantity, an acceleration occurs when an object speeds up (magnitude of elocity increases) an object slows down (magnitude of elocity decreases) an object changes direction (direction of elocity changes) Acceleration is a ector quantity. You don t sense how fast you are traelling in a car, but you do notice changes in speed and direction of the car especially if the changes occur rapidly you feel the effects of the acceleration. Some of the effects of acceleration we are familiar with include: the experience of sinking into the seat as a plane accelerates down the runway, the "flutter" in our stomach when a lift suddenly speeds up or slows down and being thrown side-ways in a car going around a corner too quickly. The aerage acceleration of an object is defined in terms of the change in elocity and the interal for the change a (5) ag definition of aerage elocity t The instantaneous acceleration is the time rate of change of the elocity, i.e., the deriatie of the elocity gies the acceleration (equation 6). Again, you don t need to differentiate a function but you need to know the notation and interpret it graphically as the acceleration is the slope of the tangent to a elocity s time graph as shown in figure (4). The area under the acceleration s time graph in the time interal is equal to the change in elocity in that time interal as shown in figure (5). (6) (7) d a definition of instantaneous acceleration dt d a rectilinear motion dt

7 elocity acceleration a elocity elocity elocity a > 0 t t a < 0 time t time t instantaneous acceleration at time t is equal to the slope of the tangent at time t a t a = 0 elocity is a linear function of time a t posite constant t Fig. 4. Velocity s time graphs for rectilinear motion. The slope of the tangent is equal to the acceleration. For the special case when the elocity is a linear function of time (straight line) the acceleration is constant. s t time t t time t time t Fig. 5. The area under the elocity s time graph in the time interal t is equal to the change in displacement s in that time interal. The area under the acceleration s time graph in the time interal t is equal to the change in elocity in that time interal.

8 VELOCITY AS A VECTOR We will consider a number of problems (with solutions) to illustrate a how-to approach to soling problems inoling ectors. It is important to consider the steps and strategies inoled so that you will use similar techniques in answering your examination questions. Remember you want to strie for technical excellence. For many ector problems you need to hae mastered the mathematics of a right angle triangle. For a right angle triangle with sides a, b and c (hypotenuse) and with angles A, B and C = 90 sin A a cos A b tan A a c c b sin A cos A 2 2 tan A sin A cos A 1 a B C b c A Pythagorean theorem Area of triangle = ½ a b c a b Example 1 An aircraft is climbing with a steady elocity of 200 m.s at an angle of 20 o to the horizontal. What are the speeds that the aircraft is moing horizontally and ertically? Solution 1 How-to approach the problem Draw a frame of reference showing the x (horizontal) and y (ertical) coordinate axes. Represent the aircraft as a dot at the origin. Draw the elocity ector Show the components of the elocity ector on a separate diagram with a box around the elocity ector. Use the properties of a right angle triangle to calculate the alues for the x and y components of the elocity. Check you answer sensible, significant figures, units. +y +x? m.s y 200 m.s 20 o = 20 o? m.s x always identifying the unknowns

9 From the right angle triangle 200cos 20 m.s 188 m.s horizontal component x ertical component y units included after all expressions, answers to 3 s.f. o 200sin 20 m.s 68.4 m.s o check: m.s 200 m.s ok x y Example 2 An aircraft is trying to fly due north with a elocity of 200 m.s but is subject to a cross wind blowing from the east at 50 m.s. What is the elocity of the plane with respect to the ground? Solution 2 How-to approach the problem Draw a frame of reference showing the x (WE) and y (SN) coordinate axes. Represent the aircraft as a dot. Draw the elocity ectors for the aircraft w.r.t. the wind and the wind w.r.t. the ground. Add the ectors using the head-to-tail method. Use the properties of a right angle triangle to calculate the resultant elocity. Check you answer sensible, significant figures, units. W N +y S +x E elocity of the plane w.r.t. the air m.s elocity of the wind w.r.t. the ground 2 50 m.s The resultant elocity of the plane w.r.t the ground is Adding the ectors head-to-tail method From the right angle triangle The resultant elocity of the plane w.r.t. the ground is 206 m.s and is flying in the direction 14.0 o W of N.? m.s m.s 206 m.s 50 atan o ? m.s always identifying the unknowns 200 o?

10 Example 3 A motor boat is traelling at 10.0 m.s w.r.t the water. The boat wants to cross from one side to the other of a 400 m wide rier in the shortest possible time but the rier is flowing at a rate of 5.00 m.s? In which direction must the boat point and how long will it take to cross the rier? Solution 3 How-to approach the problem Draw a frame of reference showing the x (perpendicular to the rier) and y (parallel to the rier) coordinate axes. Represent the boat as a dot. Draw the elocity ectors for the boat w.r.t the water and the water w.r.t. the ground. Add the ectors using the head-to-tail method so that the resultant points directly across the rier. Use the properties of a right angle triangle to calculate the elocity of the boat w.r.t. the water. Calculate the time to cross the rier. Check you answer sensible, significant figures, units. For the boat to cross the rier in the shortest time it must trael directly across the rier. So the boat must head up steam so that it will drift down with the flow of the water in the rier. +y +x water flowing in rier m.s boat w.r.t. water m.s? resultant elocity must be directly across the stream (pointing in the +x direction)? m.s o m.s From the right angle triangle o? m.s width of rier x 400 m time to cross rier t?s x x t t m.s 8.66 m.s sin asin o x 400 t s = 46.2 s 8.66 The boat must head up-stream pointed at an angle of 30 o as shown in the diagram. The boat crosses the stream at 8.66 m.s and the shortest time to cross the rier is 46.2 s.

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