Answer Happy

GMA NI a = GM 3 act) - POLOS V4) - VLAXXAS yles - OCE) OBSEIXO (CAN)2 BC) = 0 64 US X SUS VUI OGM O (COKLAT KEAYU 3 XCA GM -100 (lken KIT) HOM Koncer 2 ye) gla 4) When two vectors are equal, their components are equal (this means that when al + bj = c + dj, then a cand b=d). Equate the x and the y components on each side of the equation you got in step 3. This should give you two second-order differential equations, one involving x*(). XO), and y), and the other involving y*). x(), and 900). These equations will not have i and in them. 5) Let vx(1) = x'(), and vy(t) = y().and turn each equation into a system of two first- order differential equations. Collect these into a system of four first-order differential equations (the order should be x(i), vx(i),y(i), vy(t)). The four unknowns are the x and the y coordinate, and the velocities in the x and the y directions of each celestial object orbiting the Sun. This system of four first-order differential equations will separately model the orbits of Earth, Mars and the spacecraft, if we use the correct initial conditions for each orbiting object. In other wo if you use the four initial conditions for Mars in this system of equations, and solve them, you can get the x. and they coordinates of Mars, and its velocity in the x and the y directions for a given time.

r(t) = V(X(t))² + (y(t))2 a(t) = x"(t) i +9"(t) j = -6M (X(t))² + (ylti7² là (ty= -6M ((x(+1) + f(e)? • X(t)i + y(t) j x12? X; XH) i + g(+1] IVCE))2+(911724 4 AX"(t) = - GM • X(t) (CXCE11? 19 (11) Y"(t) = -6M -ght) (CALE + y(t)12)

4) When two vectors are equal, their components are equal (this means that when ai + bj = ci + dj , then a =c and b= d). Equate the x and the y components on each side of the equation you got in step 3. This should give you two second-order differential equations, one involving x" (t), x(t), and y(t), and the other involving y"(t), x(t), and y(t). These equations will not have i and j in them. 5) Let vx(t) = x'(t), and vy(t) = y'(t), and turn each equation into a system of two first- order differential equations. Collect these into a system of four first-order differential equations (the order should be x(t), vx(t), y(t), vy(t)). The four unknowns are the x and the y coordinate, and the velocities in the x and the y directions of each celestial object orbiting the Sun. This system of four first-order differential equations will separately model the orbits of Earth, Mars and the spacecraft, if we use the correct initial conditions for each orbiting object. In other words, if you use the four initial conditions for Mars in this system of equations, and solve them, you can get the x, and the y coordinates of Mars, and its velocity in the x and the y directions for a given time.