The solution of differential equation of physical problem by the use of Matlab

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1 The solution of differential equation of phsical problem b the use of Matlab Erika Fechoá Technical Uniersit of Košice Facult of Manufacturing Technologies of the Technical Uniersit of Košice with a seat in Prešo Department of Informatics Mathematics and Cbernetics erika.fechoa@tuke.sk Abstract At the present time teachers tr to introduce and use information technologies in educational process of technical subjects at schools as mathematics phsics chemistr. Apart from e- learning support (electronic materials and tetbooks aailable on the internet) there are arious mathematical software enironments as Matlab Mathematica MS Ecel etc. The paper points out the possibilit of Matlab utilization at soling differential equation of phsical problem. 1. Introduction Phsical problem proides knowledge in arious forms as a means of motiation information about eamined phenomenon but also as a set of functions needed for the solution of the problem. A student solutionist of phsical problem has to carr out a certain program consisting of partial functions - steps [1]. Creation of mathematical models of phsical phenomena is an important part of phsical problems solution. Modelling is one of theoretical cognitie methods whose characteristic feature is cognitie process in which eamined phenomenon an original cognitie object is replaced b a model cognitie object a model. Under the term of mathematical model is understood a sum of mathematical relations capable of quantitatie description of phsical phenomenon. Mathematical model is an abstraction and idealization of real phenomenon []. Model has to be created as a functional one. Functionalit of model is estimated according to its behaior at the change of conditions at which it works. The form of mathematical model depends on used mathematical tools. Mathematical model of phsical problem soled within the subject of Phsics of bachelor s studies at the Facult of Manufacturing Technologies of the Technical Uniersit of Košice with a seat in Prešo is presented in the following part of the paper.. The solution of phsical problem Assignment: Steel cannonball with the r radius and ρ densit is fired with the initial elocit from the cannon of the Coast Artiller under an α angle. Determine the length of o cannonball firing range. Sole the problem for numerical alues: α = 45 r = 5m = m 3 ρ 78kg. 1 = 1m. s 1 c = 6kg. m. Solution: The solution of the problem is based on the fact that the cannonball carries out curilinear motion in the graitational field of the Earth that is called projection at an angle. It

2 is the motion consisting of two motions: uniform motion and ertical throw upward with the initial elocit (fig. 1). Figure 1. The motion of cannonball. We proceed from the balance of forces to make out equations. For the balance of forces at the d d direction of ais is alid: F + F o = m + cs = where F is the power of acceleration at the direction of ais according to the Newton s second law F o is the resistance power at the direction of ais c is the coefficient of aerodnamic resistance of the cannonball S is the frontal area of the shot. For the balance of forces at the direction of d d ais is alid: F + Fo + Fg = m + cs = mg where F is the power of acceleration at the direction of ais according to the Newton s second law F g is the graitation force F o is the resistance power at the direction of ais c is the coefficient of aerodnamic resistance of the cannonball S is the frontal area of the shot. The sstem of two differential equations of the second order without or with the second member was obtained: d d d d m + cs = m + cs = mg (1) In case the resistance of enironment is not considered (i.e. the coefficient of aerodnamic resistance c = ) we hae the sstem of equations for the balance of forces at the direction of d d and aes: m = and m = mg. B their solution and integration concerning the initial conditions ( t = ) = we obtain the quadratic equation: g +. tgα = [3] from which results that trajector of motion of the cos α cannonball has the form of parabola. The solution of the quadratic equation is searched length of cannonball firing range = m. If the air resistance is not neglected at the solution of the phsical problem it is necessar to sole the sstem of differential equations of the second order (1). If the relations for the 3 weight olume and frontal area of the cannonball m = ρv 4 V = π r S = πr are used we 3

3 3 introduce the constant k = c and denote = h a = h the relations (1) will be 4ρr transposed into the form of: d h dh + k = h ( t = ) = dh = t= = cosα d h dh dh + k g h ( t ) sinα = = = = = t= The sstem of two differential equations of the second order without or with the right side of equation was obtained. An analtic solution of the problem is comple and it does not lead to such simple dependancies for the coordinates of the ball shot as it was in case the air resistance was not considered. The problem will thus be soled numericall b using an interactie program Matlab. 3. MATLAB utilization at soling phsical problem Matlab presents a highl efficient language for technical calculations. It connects calculations isualization and programming into simpl operable enironment [4]. It is an interactie deice where the basic data tpe is the field without necessit to declare its measurements. This feature together with the number of built-in functions enables relatiel eas solution of man technical problems. It is a standard tool at teaching at school for eample mathematics and other technical subjects but also an effectie tool for research deelopment and analsis of data. One of the methods of solution in Matlab is the m-files formation which were used at soling the problem. We created an m-file into which we transformed the sstem of differential equations of the second order () concerning the initial conditions into the sstem of equations of the first order in the form of matrices. Then we formed a script in which the parameters used at soling the problem were defined. Single solution consists of the calculation of length and time of the cannonball firing range and portraal of the trajector. M-file strela.m for the portraal of the distance of the cannonball firing range is created in fig.. () Figure. M-file for depiction of the graph.

4 The graph of trajector of the cannonball at gien α angle depicted b created m-file is presented in fig. 3. Figure 3. Graph of trajector. Single solution of the problem presents launching the script (fig. 4). Figure 4. M-file for the calculation of the length and time of firing range. It results from the numerical solution of such formed mathematical model of the phsical problem that firing range of the cannonball in case the air resistance is considered is = m. 4. Conclusions It emerges from the results of soling the phsical problem that the results obtained at soling the problem differ concerning the formed mathematical model. Mathematical model of the problem if the air resistance was considered led to more comple relations and numerical method of soling b Matlab was used. Introduction and utilization of information and communication means make teaching more attractie which is necessar for the subjects as mathematics phsics chemistr because the occur among the least faourite subjects.

5 Acknowledgement. This paper was supported b the grant Vega n. 1/345/8. References [1] V. Koubek. Riešenie fzikálnch úloh. MFF UK Bratislaa [] V. Koubek I. Pecen. Matematické modeloanie fzikálnch jao. Fzikálne list č [3] E. Fechoá. Matematické modeloanie ako súčasť riešenia fzikálnej úloh. In: Didmatech 7 Olomouc 7 p ISBN [4] F. Dušek. MATLAB a SIMULINK úod do použíání. Unierzita Pardubice. ISBN