DOING PHYSICS WITH MATLAB MECHANICS HORIZONTAL MOTION WITH RESISTANCE Ian Cooper School of Physics, Uniersity of Sydney ian.cooper@sydney.edu.au DOWNLOAD DIRECTORY FOR MATLAB SCRIPTS ec_fr_b. This script is used to copute the displaceent, elocity and acceleration for the otion of an object acted upon by a resistie force F. The equation of otion is soled by analytical eans (integration of the equation of otion) and by a finite difference nuerical ethod. ec_fr_b. This script is used to copute the displaceent, elocity and acceleration for the otion of an object acted upon by a resistie force F. The equation of otion is soled by analytical eans (integration of the equation of otion) and by a finite difference nuerical ethod. R R http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 1
INTRODUCTION We will consider the horizontal otion of an object of ass that is acted upon by a resistie force F R. Two ery good odels for the resistie force F R are Model (1) Model () low speeds high speeds where and are constants of proportionality. Model (1) for linear resistance is often applicable when the object is oing with low speeds. In the otion through a fluid, the resistie force is often called the iscous drag and it arises fro the cohesie forces between the layers of the fluid. The S.I. units for the constant are N. -1.s -1 or g.s -1. Model () for quadratic resistance is ore applicable for higher speeds. In the otion through fluids, the resistie force is usually called the drag and is related to the oentu transfer between the oing object and the fluid it traels through. The S.I. units for the constant are N. -.s - or g. -1. Many probles in the atheatical analysis of particles oing under the influence of resistie forces, you start with the equation of otion. To find elocities and displaceents as functions of tie you ust integrate the equation of otion. The equation of otion for a particle can be deried fro Newton s Second Law 1 (3) a Fi Newton s Second Law i When the resultant force acting on the object is siply the resistie force, the acceleration a of the object is (4) a / Model (1) a / Model () (5) http://www.physics.usyd.edu.au/teach_res/p/phoe.ht
MODEL 1 Analytical Approach The force acting on the object is the resistie force reference, we will tae to the right as the positie direction.. In our frae of The equation of otion of the object is deterined fro Newton s Second Law. d a a dt where a is the acceleration of the object at any instance. The initial conditions are / t x a We start with the equation of otion then integrate this equation where the liits of the integration are deterined by the initial conditions (t = and = ) and final conditions (t and ) d dt d dt d t dt loge t log e t (5) / t e exponential decay We can now calculate the displaceent x as a function of tie t dx dx dt dt e / t x / t dx e dt / t x e t / t (6) x 1 e http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 3
The elocity also can be gien as a function of x d d a dt dx d dx x d dx (7) x x straight line graphs When = the stopping distance is xstopping We can now inestigate what happens when t t e / t t x 1 e / t The object eeps oing till =, which happens only in the liit t. Then the object stop at the position x. http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 4
We can define a tie constant The elocity and displaceent can be expressed as e t / x 1 e t / After a tie of about 5, the particle will stop when the speed of the particle becoes zero x stopping stopping tie ~ 5 The stopping tie is independent of the initial elocity but the greater the initial elocity the greater the stopping distance. The larger the constant, the shorter the stopping distance and quicer it stops and the larger the ass, the greater the stopping distance and it taes a longer tie to stop the object. Nuerical Approach We can also find the elocity and displaceent of the object by soling Newton s Second Law of otion using a finite difference ethod. We start with d dt (4) a / In the finite difference ethod we calculate the elocity and displaceent x at N discrete ties t at fixed tie interals t t 1, t,, t,, t N t =t - t 1 t 1 = t = (-1) t = 1,, 3,, N The acceleration a is approxiated by the difference forula d t t t dt t 1 http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 5
t Therefore, the elocity t t t at tie t + is / t 1 / t t t t 1 t Hence, to calculate the elocity we need to now the elocity at the two preious tie steps t 1 and t. We now t 1 = and (t 1 ) = () =. We estiate the elocity at the second tie step t (8) ( ) ( ) ( ) ( ) / t t a t t t t t 1 1 1 1 where we hae assued a constant acceleration in the first tie step. We can iproe our estiate of t ( ) by using an aerage alue of the acceleration in the first tie step a( t1) a( t) ( t1) t ( t) 1 ( t1) / t1 t t ( t) We can now calculate the elocity (t) at all ties fro t = t 1 to t = t N. The acceleration at each tie step is (9) a t t The displaceent at each tie step is dx x dt t x t t 1 x t t (1) xt xt tt 1 http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 6
EXAMPLE The script ec_fr_b. can be used for siulations for the otion of an object acted upon a resistie force of the for Input paraeters = g = 1 g.s -1 = 1.s -1 t = 1x1-4 s Outputs N nuerical approach A analytical approach (Model 1). stopping distance x stopping =. tie constant =. s 5 = 1. s acceleration t a elocity t http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 7
displaceent t x xstopping For the input paraeters used in this siulation, there is excellent agreeent between the alues calculated using the nuerical and analytical approaches. Howeer, you always need to be careful in using nuerical approaches to sole probles. In this instance, you need to chec the conergence of results by progressiely aing the tie step t saller. When the tie step is t = 1x1 - s the nuerical and analytical results do not agree. The tie step is too large for accurate results using the nuerical approach. http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 8
MODEL Analytical Approach The force acting on the object is the resistie force reference, we will tae to the right as the positie direction.. In our frae of The equation of otion of the object is deterined fro Newton s Second Law. d a a dt where a is the acceleration of the object at any instance. t x a / The initial conditions are We start with the equation of otion then integrate this equation where the liits of the integration are deterined by the initial conditions (t = and = ) and final conditions (t and ) d dt d dt t d dt 1 1 1 t 1 t (11) 1 The acceleration is t (1) a 1 1 t 1 t t a acceleration becoes sall ery rapidly http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 9
We can now calculate the displaceent x as a function of tie t d d a / dt dx d dx x d dx x log (13) x log (11) 1 t (14) x log 1 t surprising result!!! As tie goes on the displaceent gets bigger and bigger. In this siple odel, the objects just eeps oing towards the right. Nuerical Approach We can also find the elocity and displaceent of the object by soling Newton s Second Law of otion using a finite difference ethod. We start with d a dt / (4) In the finite difference ethod we calculate the elocity and displaceent x at N discrete ties t at fixed tie interals t t 1, t,, t,, t N t =t - t 1 t 1 = t = (-1) t = 1,, 3,, N The acceleration a is approxiated by the difference forula d t t t dt t 1 http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 1
t Therefore, the elocity t t t at tie t + is / t 1 / t t t t 1 t Hence, to calculate the elocity we need to now the elocity at the two preious tie steps t 1 and t. We now t 1 = and (t 1 ) = () =. We estiate the elocity at the second tie step t (8) ( ) ( ) ( ) ( ) / t t a t t t t t 1 1 1 1 where we hae assued a constant acceleration in the first tie step. We can iproe our estiate of t ( ) by using an aerage alue of the acceleration in the first tie step a( t1) a( t) ( t1) t ( t) 1 ( t1) / t1 t t ( t) We can now calculate the elocity (t) at all ties fro t = t 1 to t = t N. The acceleration at each tie step is (9) a t t The displaceent at each tie step is dx x dt t x t t 1 x t t (1) xt xt tt 1 http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 11
EXAMPLE The script ec_fr_b. can be used for siulations for the otion of an object acted upon a resistie force of the for Input paraeters (Model ). = g = 5 g. -1 = 1.s -1 t = 1x1-4 s t ax = 5 s Outputs N nuerical approach A analytical approach acceleration t a acceleration quicer gets saller in agnitude with tie elocity t http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 1
displaceent does not go to zero as t For the input paraeters used in this siulation, there is excellent agreeent between the alues calculated using the nuerical and analytical approaches. Howeer, you always need to be careful in using nuerical approaches to sole probles. In this instance, you need to chec the conergence of results by progressiely aing the tie step t saller. When the tie step is t = 1x1-3 s the nuerical and analytical results do not agree. The tie step is too large for accurate results using the nuerical approach. In the nuerical approach the position alues oscillate about the analytical alues. http://www.physics.usyd.edu.au/teach_res/p/phoe.ht 13