Effects of Projectile Motion in a Non-Uniform Gravitational Field, with Linearly Varying Air Density

Transcription

1 Effects of Pojectile Motion in a Non-Unifom Gavitational Field, with Linealy Vaying Ai Density Todd Cutche Decembe 2, 2002 Abstact In ode to study Pojectile Motion one needs to have a good woking model of how a pojectile is effected by extenal foces. The model in this pape will exploe the effects of vaiable gavity, vaiable ai esistance, and cuvatue of the eath, on pojectile motion.

2 Intoduction To study Pojectile Motion the fist thing that we need to is come up with a good model of how a pojectile tavels. To this popely we need to account fo (among othe things not coveed hee) the effects of: vaiable gavity, vaiable ai esistance, and the cuvatue of the eath. Once we have found an expession fo the esistive foce, (this will be the foce opposing the pojectile due to gavity and ai esistance) we will use it to get a set of fou dynamical equations. This set of dynamical equations will be solved numeically fo the path of the pojectile. The numeic solutions to these dynamical equations will be pefomed using the fouth ode Runge-Kutta method pogammed in C++ (see Appendix A). 2 Theoy To come up with a good model fo pojectile motion, we must fist deive a set of dynamical equations that can be solved (eithe analytically o numeically) fo the path of the pojectile. These equations have to account fo:. Launching a pojectile at any angle (0 o 90 o ). 2. The cuvatue of the eath. 3. The fact that gavity vaies with the height of an object. 4. The fact that ai esistance vaies with the height of an object. To stat off we will get an expession fo the foce of a gavity field that is dependent on altitude. This will be ne by using Newton s Law of Univesal Gavitation: F GMm R 2 u () whee G 6 672x0 Nm 2 and is the univesal gavitation constant. M is the mass of the intestella body kg 2 that you ae on (in ou case M Eath 5 98x0 24 kg), m is the mass of the object being thown, and R is the adial distance measued between the cente mass of the two objects. We next need to get an expession fo ai esistance that is dependent on the altitude of ou pojectile (). To this we used the book s quadatic model of ai esistance ( F c v v ) whee c 22D 2 and D is the diamete of the pojectile. To get a vaiable ai esistance we make the assumption that this is coect at sea level ( ) only, and that the ai esistance educes linealy to zeo at the edge of the atmosphee (R a ). In ode to accomplish this we need to multiply this equation by some expession that is one at and zeo at R a. This leads us to the expession: F c Ra v v (2) By combining equations () and (2) we ae able to get the following expession fo the foces opposing the motion of ou pojectile. F GMm 2 u c R a v v (3) By dividing this equation by m and by beaking v v into its component vectos we get. 2

3 ā GM c 2 u m GM c 2 u m c m R a Ra R a v φ v 2 v φ 2 v v (4) v v 2 v φ2 u u φ (5) We also know fom class 2, and the Round Eath and Vaiable Gavity Effects on Pojectile Motion hanut (equation # (.20)), that ā 2 φ u 2ṙ φ φ u φ z u z (6) By combining equations (5) and (6) we ae able to get: φ 2 GM 2 c m Ra v v 2 v φ 2 (7) 2ṙ φ φ c m and Ra v φ v 2 v φ 2 (8) Finally by using equations (7) and (8) we ae able to find the dynamical equations. This is ne by solving each equation fo and φ espectively. d dt dṙ dt dφ dt d φ dt ṙ (9) φ 2 GM c Ra 2 m R e v v 2 v 2 φ (0) φ () 2ṙ φ c m R a v φ v 2 v φ 2 (2) These ae fou coupled second ode odinay diffeential equations and theefoe a numeic solution is equied. 3

4 3 The Pogam To solve the dynamical equations (9, 0,, & 2) I modified the C++ Runge-Kutta pogam that was povided in class 2. This pogam uses fouth ode Runge-Kutta to numeically solve diffeential equations. The pogam was changed to solve the dynamical equations fo and φ with pojectiles of any diamete (D), initial velocity (V o ), and launch angle (θ) this pogam was also setup to take these initial values as command line aguments. Taking the initial values as command line aguments allowed the Pojectile Launch pogam (Appendix A) to be un by a bash scipt (Appendix B) that would pass it the initial values. The pogam was then un fo objects with a diamete of 0.0 m,.0 m, 0.0 m, & 0.00 m. The initial velocity and launch angle vaied fom 0.0 m/s to 0000 m/s (in incements of 0 m/s) and 0 o to 90 o (in incements of 5 o ) espectively. This esulted in data sets. Note: The density used fo all objects is that of a baseball ( kg/m 3 ), & all pojectiles ae launched fom km above the Eath s suface. 4 Results The esults fo all the data sets tun out to be some what simila. Because of these similaities, and shee volume, I am not including all of the data. Seveal examples of data sets ae included in Appendix C. In evey case at high initial velocity the object is able to escape, at low initial velocity the object etuns to the Eath. An inteesting point is that no object was found to be able to cicle the Eath. This is because the launch position is only km above the eath suface, and so the effect of ai esistance is still quite lage. 4

5 Figue : Pojectiles with D 0m, θ 40 o, & vaying initial velocity s. In Figue it can be seen that fo high velocities such as the geen and bown taces (V o 0000 m s & V o 5000 m s) that the pojectile will escape the Eath s gavitational pull. and fo lowe initial velocity s such as the ed, blue, and black taces (V o 000 m s & V o 500 m s& V o 00 m s) the pojectile will eventually etun to Eath. This figue shows what is typical fo all of the data collected. Figue 2: Pojectiles with D m,v o 000 m s and vaying initial velocity s θ 5

6 Figue 2 shows an example of the compaison between diffeent initial launch angles fo pojectiles with the same diamete and velocity. The taces on this figue ae fo launch angles of 5 o to 90 o in incements of 5 o. The diamete and velocity of these pojectiles is m and 000 m/s espectively. This figue also shows (as was expected) that the optimum launch angle to achieve maximum ange is appoximately 45 o. 5 Conclusion The esults obtained fom this model of pojectile motion ae what would be expected of a good model. If you launch a pojectile with an extemely lage initial velocity it will escape the gavitational pull of the Eath. The angle that you launch it at has an effect on the flight of the pojectile. The optimum launch angle is appoximately 45 o. I was not able to show that you could put an object into obit, howeve this is due to the launch position of the pojectile. 6

7 6 Appendix A The following code is the C++ pogam used fo this poject. #include <iosteam.h> #include <stdio.h> #include <math.h> #define N 4 // numbe of fist ode equations #define DELTA_T 0.09 // stepsize in t #define T_MAX // max fo t #define GM e4 #define R_e //ad. of eath (m) #define R_a //distance fom cente of eath to space (m) #define Dens //density of a baseball (kg/mˆ3) #define Pi static float c; static float m; int main(int agc, cha **agv){ float Diam; //diam. of object (m) float V_o; //inital velocity (m/s) float theta_o_deg; //launch angle (deg.) uble t, y[20],inputs[agc]; int i,j,k; void unge4(uble x, uble y[], uble step); // Runge-Kutta function if (agc<=3){ cout << "\n\n\nyou need to include command line aguments!!\n" << endl; cout << "\nexample:./a.out <Diam> <V_o> <theta_o_deg>!!\n\n\n" << endl; etun 0; } else{ sscanf(agv[], "%f", &Diam); sscanf(agv[2], "%f", &V_o); sscanf(agv[3], "%f", &theta_o_deg); } float theta_o_ad = theta_o_deg*(pi/80); //launch angle (ad.) c = 0.22*Diam*Diam; m = Dens*(4.0/3.0)*Pi*(Diam/2)*(Diam/2)*(Diam/2); y[0]= R_e+; // initial y[]= V_o*sin(theta_o_ad); // initial _t y[2]= 0.0; // initial phi y[3]= V_o*cos(theta_o_ad)/R_e; // initial phi_t pintf("%lf\t", y[2]); pintf("%lf\t", y[0]); 7

8 pintf("\n"); fo (j=; j*delta_t <= T_MAX ;++j) { t=j*delta_t; unge4(t, y, DELTA_T); // time loop if (y[0]<=r_e)etun 0; // will check if the pojectile has hit the gound pintf("%lf\t", y[2]); pintf("%lf\t", y[0]); pintf("\n"); } etun 0; }//end of main uble f(uble x, uble y[], int i) { uble V_ = y[]; uble V_phi = y[0]*y[3]; if (i==0) etun(y[]); if (i==) etun((y[0]*y[3]*y[3])-(gm/(y[0]*y[0]))-((c/m)*((-(y[0]/r_a))/(-(r_e/r_a))) *(V_*sqt((V_*V_)+(V_phi*V_phi))))); if (i==2) etun(y[3]); if (i==3) etun(((-2*y[]*y[3])/y[0])-((c/(m*y[0]))*((-(y[0]/r_a))/(-(r_e/r_a))) *(V_phi*sqt((V_*V_)+(V_phi*V_phi))))); }//end of f void unge4(uble x, uble y[], uble step) { uble h=step/2.0; // the midpoint uble t[n], t2[n], t3[n]; // tempoay stoage aays uble k[n], k2[n], k3[n],k4[n]; // fo Runge-Kutta int i; fo (i=0;i<n;i++) t[i]=y[i]+0.5*(k[i]=step*f(x, y, i)); fo (i=0;i<n;i++) t2[i]=y[i]+0.5*(k2[i]=step*f(x+h, t, i)); fo (i=0;i<n;i++) t3[i]=y[i]+(k3[i]=step*f(x+h, t2, i)); fo (i=0;i<n;i++) k4[i]=step*f(x+step, t3, i); fo (i=0;i<n;i++) y[i]+=(k[i]+2*k2[i]+2*k3[i]+k4[i])/6.0; etun; }//end of unge4 8

9 7 Appendix B Below is the bash scipt used to un the C++ pogam #!/bin/bash Diam=0.00 V_o=0 while [ "$V_o" -lt 000 ] # Beginning of oute loop theda=0 # Reset theda. while [ "$theda" -lt 9 ] # Beginning of inne loop./theo_poj_.2.out $Diam $V_o $theda > $Diam"_"$V_o"_"$theda.dat let "theda += 5" ne # End of inne loop let "V_o += 0" ne # Incement pass count. # End of oute loop # All ne. Diam=0.0 V_o=0 while [ "$V_o" -lt 000 ] # Beginning of oute loop theda=0 # Reset theda. while [ "$theda" -lt 9 ] # Beginning of inne loop./theo_poj_.2.out $Diam $V_o $theda > $Diam"_"$V_o"_"$theda.dat let "theda += 5" ne # End of inne loop let "V_o += 0" ne # Incement pass count. # End of oute loop # All ne. 9

10 Diam=.0 V_o=0 while [ "$V_o" -lt 000 ] # Beginning of oute loop theda=0 # Reset theda. while [ "$theda" -lt 9 ] # Beginning of inne loop./theo_poj_.2.out $Diam $V_o $theda > $Diam"_"$V_o"_"$theda.dat let "theda += 5" ne # End of inne loop let "V_o += 0" ne # Incement pass count. # End of oute loop # All ne. Diam=0.0 V_o=0 while [ "$V_o" -lt 000 ] # Beginning of oute loop theda=0 # Reset theda. while [ "$theda" -lt 9 ] # Beginning of inne loop./theo_poj_.2.out $Diam $V_o $theda > $Diam"_"$V_o"_"$theda.dat let "theda += 5" ne # End of inne loop let "V_o += 0" ne # Incement pass count. # End of oute loop # All ne. 0

11 8 Appendix C Figue 3: Pojectiles with D 0m, θ 75 o, & vaying initial velocity s. Figue 4: Pojectiles with D m, θ 0 o, & vaying initial velocity s.

12 Figue 5: Pojectiles with D 0 0m, θ 40 o, & vaying initial velocity s. Figue 6: Pojectiles with D 0 00m, θ 75 o, & vaying initial velocity s. 2

13 9 Refeences. Fowles and Cassiday Analytical Mechanics, sixth edition 2. Class notes & hanuts (Tutoial s) D. Anthony A. Tova Easten Oegon Univesity Theoetical Physics Pogamming and Poblem Solving with C++ Nell Dale Chip Weems Mak Headington 3