Vectors, elocit and displacement Sample Modelling Actiities with Excel and Modellus ITforUS (Information Technolog for Understanding Science) 2007 IT for US - The project is funded with support from the European Commission 119001-CP-1-2004-1-PL-COMENIUS-C21. This publication reflects the iews onl of the author, and the Commission cannot be held responsible for an use of the information contained therein.
2 I. Introduction These set of actiities illustrates how a modelling program like Modellus can be used to illustrate phsical ector quantities and how these tpe of quantities relate to scalar quantities. 1. Background Theor Phsical quantities can be diided into two categories: scalars and ectors. While ector quantities to be full described need both magnitude and direction, scalar quantities onl need a real number. For most common ector phsical quantities, it is possible to describe magnitude and direction geometricall using arrows, a segment with a tail and a tip. The length of the arrow represents magnitude (it then needs a scale!) and the tip represents the direction. Displacement, elocit, acceleration, force are examples of ector quantities. tail tip Distance is an example of a scalar quantit, in this case bigger or equal to zero since it is a numerical description of how far apart objects are at a certain moment in time. Temperature in the Kelin scale is another example of a similar phsical quantit. The direction of a ector can be expressed as an angle of rotation of the ector about its tail. For example, once defined a certain direction, such as the northern direction, a ector can hae a direction of 30 degrees west of north: that ector has been rotated 30 degrees towards the westerl direction. The most common conention is to express the direction of a ector as a counter clockwise angle of rotation about its tail from north due east. Then, a ector with a direction of 90 degrees is a ector rotated 90 degrees in a counter clockwise direction from north relatie to east. And a ector with a direction of 180 degrees has been rotated 180 degrees: it points south. Etc. In naigation, directions are usuall expressed with three digits (from 000 to 360), as shown below. to north direction of 30 degrees west of north 30º to west 315 degrees 270 degrees west 000 degrees (or 360) to north 180 degrees south 045 degrees 090 degrees east A B The distance between A and B is a positie real number (a scalar): it measures the lenght of the straight line between A and B. The lenght of the trajector (represented in black) is also a positie scalar. The displacement from A to B is a ector: its magnitude is equal to the distance betwenn A and B and its direction sas that the motion started in A and finished in B
3 A ector can be multiplied b a scalar. When such multiplication is done, the new ector has the same direction if the scalar is positie or the opposite direction if the scalar is negatie. Multipling elocit (a ector) b time interal (a positie scalar) gies displacement since elocit is the instantaneous rate of change of position (displacement). For constant elocit, we hae Dr = Dt Dr = Dt displacement = elocit time interal If the elocit is not constant oer a certain time interal, one needs to consider small time interals where it can be considered constant (in realit or just as an approximation). In this conditions, the total displacement is the sum of the displacements in each small time interal: total displacement D r = Dt 4 4 4 = Dr + Dr + Dr + D Dr r 1 2 3 3 D r = Dt 2 2 2 D r = Dt 3 3 3 D r = Dt 1 1 1 Vector addition can be made placing the tail of one ector on the tip of the other: the ector sum connects the free tail to the free tip. This rule can be used successiel, as shown on the picture. In a plane, a ector can be represented b the sum of two ector components, one on the x axis and the other on the axis. For each ector component, there is the correspondent scalar component, a real number. Four examples: = 3 O = 2 x = 3 O =-3 x a ector... the magnitude of the ector... 2 2 x = + the ector component on the x axis =-1 O =-4 x = 0 = 0 O º =-3 x the ector component on the axis the scalar component of the ector on the axis (a real number) the scalar component of the ector on the x axis (a real number)
4 Scalar components of a ector can be expressed using trigonometric functions, as shown below: q q -180º O x = cos() q = cos q -180º sin 270º q x x x ( ) = ( - ) 270º -q = sin() q = sin q -180º cos 270º ( ) = ( -q) This Module uses the idea of numerical solution of the equations of motion. Let s see how this idea works... If we know the rate of change of quantit, which b definition is expressed b change in the quantit rate of change of a quantit = time taken to change then we can write: change rate of change of = in the quantit the quantit time taken to change Since the change of a quantit can also be expressed as change new alue of = - in the quantit the quantit preious alue of the quantit we get, combining these two last equations: new alue of the quantit preious alue of - = the quantit rate of change of the quantit time taken to change new alue of the quantit preious alue of = + the quantit rate of change of the quantit time taken to change Using smbols and considering elocit as the rate of change of position, the aboe reasoning can be represented as: Dr = D t Dr = Dt Dr = rt+dt -rt rt+dt - rt = Dt r = r + Dt t+dt t Using components, we hae: x = x + Dt t+dt t x = + Dt t+dt t
5 2. Science concepts introduced in this module Vector and scalar quantities Vector components Vector addition Position Velocit Trajector Distance traelled Rates of change Iteratie solution of the equations of motion 3. Other information The actiities concentrate on modelling motion without using a force law, i.e., an expression to compute the magnitude of the force. Learners onl use elocit to compute next position. This is particular useful where the force law is not know or irreleant, such as in naigation, in sea, land and air. In the last actiit learners are inited to make a model using acceleration. In this model, elocit in instant t + t is computed from acceleration in instant t and time interal. And acceleration can be computed from force using Newton s fundamental law of motion. Models with acceleration can be er useful to illustrate the mathematical meaning of inertia: acceleration change elocit... anf if acceleration if zero, elocit doesn t change...
6 II. Didactical approach 1. Pedagogical context The actiities presented in this module can be used with students of upper secondar school, first ear college students and secondar teachers, either in Phsics or in Mathematics classes. The were not designed to fit in an curriculum. The simple illustrate how two interactie computer tools (Modellus and a spreadsheet like Excel) can be used to model phsical phenomena. The can be particularl useful for simultaneous training of Phsics and Mathematics teachers, promoting interdisciplinarit and reflection about concepts and representations and for the introduction of simple numerical methods. 2. Common student difficulties Some of the student difficulties include: Interpreting graphs with time as the independent ariable plotted on the horizontal axis. Using rates of change to define equialent iteratie equations. Computing ector components. 3. Ealuation of ICT Computers are now the most common scientific tool, used in almost all aspects of the scientific endeaour, from measuring and modelling to writing and snchronous communication. It should then be natural to use computers in learning science. Tools like Modellus or spreadsheets are particularl useful when large number of repetitie calculations need to be made, such as ariables that are incrementall changed, and data must be presented graphicall. In the case of Modellus, learners can also use ectors to represent rates of change and interactiel see how these rates of change affect the alues of phsical quantities. Using both Modellus and Excel allows the learner to compare how different programs implement the same mathematical idea. This help learners to get familiar with the mathematical idea, not onl the specific sntax of a particular program.
7 4. Teaching approaches Good classroom organization is an essential component in a successful teaching approach, particularl when using complex tools such as computers and software. Most approaches to classroom organization that can gie good results mix features of students autonomous work, both indiiduall and in small groups, to teacher lecturing to all class. Tpicall, teachers can start with an all class approach, with students following the lesson with a screen projector. It is almost alwas a good idea to ask one or more students to work on the computer connected to the projector. This allows the teacher to hae direct information of students difficulties when manipulating the software and to be slower on the explanation of the ideas and actiities that are being presented. As all teachers know b experience, it is usuall difficult for most students to follow written instructions, een when these instructions are onl a few sentences long. To oercome this difficult, teachers can ask students to read the actiities before starting them and then promote a collectie or group discussion about what is supposed to be done with the computer. As a rule of thumb, students should onl start an actiit when the know what the will do on the actiit: the will onl consult the written worksheet just for checking details, not for following instructions.
8 III. Actiities Where does it go? The first actiit is a isual explanation about the meaning of two ector quantities (elocit and displacement) and the relation between them. It is important to get learners familiar with nautical miles and knots. If necessar, the tutor can mention that a nautical mile is simpl a minute of arc (1/60 of 1/90 of the arc between the equator and the pole) along a meridian of the Earth. The final point will be different... but the distance traelled will be the same! Distance traelled and displacement are two different phsical quantities. Direction can be represented in different was. Traditionall, in naigation, people use angles measured from the top, increasing in the same direction as a hand clock. Learners can practise how familiar the are with the correspondence between the direction measured b the angle and the direction expressed using cardinal points. Learnes should be encouraged to use a few times the ector equation displacament = elocit time interal and express the quantities as ectors in a correct scale. On this second page learners can explore how to compute sucessie displacements with different elocities (different in magnitude and, or, different in direction). Carefull exploration of this page illustrates how does an iteratie process works: the new alue of position = old alue of position + change, where change is displacement... and displacement is computed as the product of the rate of change of position (i.e., elocit) times time interal. Learners can explore different ectors and different time interals, building similar tables.
9 A model of the motion of the boat These two actiities guide learners to build a simple iteratie model in Modellus of a boat moing with a constant speed of 10 units in a direction that points nordest (if north points to the top of the page). Look carefull at the text of the model and of the initial conditions and the initial alues of the parameters. The Control window needs to be changed (change the Max alue of steps as well as the Model Tpe to iteratie). Learners should be encouraged to analse other alues for the initial alues and een where is the final position of the boat for these different alues. This actiit makes the iteratie model of the motion of the boat more interactie (some learners tend to confound iteratie with interactie; assure ourself that this does not happens!). The direction of the elocit is now changeable, using a bar that can change the angle from 360º to +360º. Learners can tr different trajectories, incluinding circular trajectories (or almost circular...). A possible extension of this actiit is adding another bar for, the magnitude of elocit, on the Animation Window. Using the two bars, one for the angle and the other for speed, learners can change interactiel both properties of elocit, direction and magnitude. It also possible to change the model to use a ector to interactiel change both properties of elocit. The image on the right shows a simple model that allows the user to change the elocit with the mouse. Tr it now... or wait for the last page of this module where this is illustrated in more detail.
10 Iteratie solution of the equations of motion These three pages snthesize the meaning of an iteratie solution of the equations of motion, explains how the can be implemented in a spreadsheet and how an iteratie solution is related with functions. Probabl, it is wise not to build the last model (page 7) since one must be er careful in order to make no mistakes... It should be enough to analse it in detail and discuss how time is delaed in each function. A somewhat challenging project could be implementing that model on the same spreadsheet of the iteratie model (page 6)! This first page snthesize how the iteratie solution works in a two dimensional space and how the iteration is expressed both in ector notation and using scalar components in two axes. This detailed explanation of how to build the spreadsheet must be carefull analsed and followed. Most spreadsheet users don t know basic mathematical sntax on a spreadsheet, such as writing the number, and how to use a factor to change from degrees to radians. Check if that is not the case of the learners using this actiit... This graph is NOT the graph of a function. It is a trajector in the Ox plane. And, since it represents point on space, it must hae equal scales on both axes. Is necessar, make also the graph of the alues of x and as functions of t. Learners can be inited to sketch this two graphs using the alues on the table and discuss how the are difficult to interpret, since the trajector is not a straight line. It could also be interesting to make the graph of total distance traelled as a function of time (see below). distance traelled ertical coordinate t/s horizontal coordinate x t/s
11 A simple and powerful model... This final actiit can be used to pla with all the concepts discussed aboe. Veclocit is now represented as an independent ector that can be freel changed b the learner, both in magnitude and direction. Analse how to get different trajectories, as the one shown (almost circular!). An interesting discussion with this model can be about how these tpe of models can be used to create computer games. It is important to note that this model doesn t take inertia into account. This means that elocit changes instantaneousl, what is not true for real moing objects. The model below presents a more correct model that takes shows how inertia can be modelled mathematicall. In this more complete model, each elocit component depends on the respectie acceleration component. The independent ector is now acceleration. Changing acceleration, both in magnitude and direction, moes the particle. Making the acceleration equal to 0, the particle continues moing, if it was moing... and remains stoped if it was stoped. Don t gie big alues to the magnitude of the acceleration... (is necessar, change the scale of the acceleration ector to a small scale, such as 1 pixel = 0.1 units).