The lab report should follow the format shown in Report Template in Content/Assignments/Computer Project/folder on D2L....

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The lab report should follow the format shown in Report Template in Content/Assignments/Computer Project/folder on D2L. Numerical results should be organized and put into Section 3 NUMERICAL RESULTS. For this report, there should be at least five figures: one for (b), three for (c) and one for (e). The program is similar to labo.m posted on D2L. For this particular lab, the following code is provided for your reference. You are to enter the result from (a) to the G array in function planets. function [t,x] = labl (V20) 8 V20 is the user-entered initial speed of mass m2 G-6.67408e-11; m1 6.4151e23; m2=3530e1; Gm1=G*m1; Gm2=G*m2; 8 gravitational constant 8 mass of Mars 8 mass of Viking Explorer I or heavy obj (e) 8 Initial Conditions at point A arranged as column vector 8 rl [x1 y1]; r2=[x2;y2]; v1=d/dt r1; v2-d/dt r2 r1 [0,0]; r2 [10] 11.1e6; v1=[0;0]; v2 [0;1]*V20; dt=1; npt 1000000; tspan [1:npt]*dt; opt odeset('RelTol',le-9, 'AbsTol', le-9); q0=[r1;12;v1;v2]; [t,x]=ode 45(@planets, tspan, q0, opt, Gm1,Gm2); 8 sampling time interval 8 number of data points 8 time period for simulation 8 error control for integration 8 initial conditions * plotting the Mars circle with radius 3389.5e3 [m] viscircles([0 0],3389.5e3, 'color','b'); hold on; 8 plotting the trajectory of Viking Explorer I plot(x(1,3),x(1,4),'r-'); function G = planets (t,q, Gm1, Gm2) 8 (x1,y1,x2,y2) (q(1), (2), q(3), q(4)) 8 d(x1,y1,x2,y2)/dt = (q(5), q(6), q(7), q(8)) R3 (sum (q(3:4)-q(1:2)).^2]].^1.5; G=[0;0;0;0;0;0;0;0]; Celestial bodies subjected to gravitational force are central force system; see Figure 1. Dynamics of the central force system is of great importance in many branches of science and engineering. In mechanics, the understanding of the path of satellite, asteroid to the evolution of stars are all related to the central force system dynamics. According to the Newton's law, the gravitational force is given by F = GMM/R where G = 6.67408 10-11 [m kg/s] is the gravitational constant, M, M are the mass of the bodies, and R is the distance between the bodies. The equations of motion for two bodies under gravitational force are given by (1) MMz M = G (1-1) R M = G MM R (-12) Figure 1 F where R = | - |, and r = x+ yj, = x2 + y2j are the position vectors for M1, M2, respectively. Consider the Viking Explorer I orbiting the planet Mars in 1976. Let M = 6.4139 1023 [kg] be the estimated mass of the Mars and M = 3530 [kg] be the mass of the Viking Explorer I. Assume the gravitational force from other celestial bodies in the Universe is negligible. Answer the following questions. (a) Write down the equations of motion in the first order form in G, M, M, R, x, y, x (b) Assume Viking travels at the speed of 2.7 103 [m/s] at A, at a distance d = 11.1 106 [m] from the Mars' center; see Figure 2. Consider initial conditions x(0) = y(0) = 0, x(0) = y(0) = 0 for Mars, and initial conditions x2(0) = 11.1 106, y(0) = 0, x(0) = 0, y(0) = 2.7 103 for the Viking. Simulate the dynamics of the system and show the orbit of the Viking and Mars. (c) To change the orbit, Viking will decrease its speed at A; see Figure 2. Follow (b) with the same initial conditions but a smaller y(0). Experiment the speed from 2.7778 103 [m/s] to 1 102 [m/s] to identify (cl) the speed at A that allows the Viking to circle the Mars with approximately the radius 11.1 106 [m] from the center around Mars, (c2) the minimum speed that Viking must maintain above at A before it crashes onto the surface of Mars, and (3) the critical speed at A for the Viking to escape from the Mars' gravitational field. (d) Research the literature and find the theoretical results for questions (c). Compare your numerical results with the theoretical values. Figure 2 B Xx A d (e) Instead of the small Explorer, imagine a heavy object 30% the mass of Mars in (b). Note, in this case, Mars will move away from its regular orbit. So you should plot the center of the Mars and the center of the object. The lab report should follow the format shown in Report Template in Content/Assignments/Computer Project/folder on D2L. Numerical results should be organized and put into Section 3 NUMERICAL RESULTS. For this report, there should be at least five figures: one for (b), three for (c) and one for (e). The program is similar to labo.m posted on D2L. For this particular lab, the following code is provided for your reference. You are to enter the result from (a) to the G array in function planets. function [t,x] = labl (V20) 8 V20 is the user-entered initial speed of mass m2 G-6.67408e-11; m1 6.4151e23; m2=3530e1; Gm1=G*m1; Gm2=G*m2; 8 gravitational constant 8 mass of Mars 8 mass of Viking Explorer I or heavy obj (e) 8 Initial Conditions at point A arranged as column vector 8 rl [x1 y1]; r2=[x2;y2]; v1=d/dt r1; v2-d/dt r2 r1 [0,0]; r2 [10] 11.1e6; v1=[0;0]; v2 [0;1]*V20; dt=1; npt 1000000; tspan [1:npt]*dt; opt odeset('RelTol',le-9, 'AbsTol', le-9); q0=[r1;12;v1;v2]; [t,x]=ode 45(@planets, tspan, q0, opt, Gm1,Gm2); 8 sampling time interval 8 number of data points 8 time period for simulation 8 error control for integration 8 initial conditions * plotting the Mars circle with radius 3389.5e3 [m] viscircles([0 0],3389.5e3, 'color','b'); hold on; 8 plotting the trajectory of Viking Explorer I plot(x(1,3),x(1,4),'r-'); function G = planets (t,q, Gm1, Gm2) 8 (x1,y1,x2,y2) (q(1), (2), q(3), q(4)) 8 d(x1,y1,x2,y2)/dt = (q(5), q(6), q(7), q(8)) R3 (sum (q(3:4)-q(1:2)).^2]].^1.5; G=[0;0;0;0;0;0;0;0]; Celestial bodies subjected to gravitational force are central force system; see Figure 1. Dynamics of the central force system is of great importance in many branches of science and engineering. In mechanics, the understanding of the path of satellite, asteroid to the evolution of stars are all related to the central force system dynamics. According to the Newton's law, the gravitational force is given by F = GMM/R where G = 6.67408 10-11 [m kg/s] is the gravitational constant, M, M are the mass of the bodies, and R is the distance between the bodies. The equations of motion for two bodies under gravitational force are given by (1) MMz M = G (1-1) R M = G MM R (-12) Figure 1 F where R = | - |, and r = x+ yj, = x2 + y2j are the position vectors for M1, M2, respectively. Consider the Viking Explorer I orbiting the planet Mars in 1976. Let M = 6.4139 1023 [kg] be the estimated mass of the Mars and M = 3530 [kg] be the mass of the Viking Explorer I. Assume the gravitational force from other celestial bodies in the Universe is negligible. Answer the following questions. (a) Write down the equations of motion in the first order form in G, M, M, R, x, y, x (b) Assume Viking travels at the speed of 2.7 103 [m/s] at A, at a distance d = 11.1 106 [m] from the Mars' center; see Figure 2. Consider initial conditions x(0) = y(0) = 0, x(0) = y(0) = 0 for Mars, and initial conditions x2(0) = 11.1 106, y(0) = 0, x(0) = 0, y(0) = 2.7 103 for the Viking. Simulate the dynamics of the system and show the orbit of the Viking and Mars. (c) To change the orbit, Viking will decrease its speed at A; see Figure 2. Follow (b) with the same initial conditions but a smaller y(0). Experiment the speed from 2.7778 103 [m/s] to 1 102 [m/s] to identify (cl) the speed at A that allows the Viking to circle the Mars with approximately the radius 11.1 106 [m] from the center around Mars, (c2) the minimum speed that Viking must maintain above at A before it crashes onto the surface of Mars, and (3) the critical speed at A for the Viking to escape from the Mars' gravitational field. (d) Research the literature and find the theoretical results for questions (c). Compare your numerical results with the theoretical values. Figure 2 B Xx A d (e) Instead of the small Explorer, imagine a heavy object 30% the mass of Mars in (b). Note, in this case, Mars will move away from its regular orbit. So you should plot the center of the Mars and the center of the object.