The Clohessey-Wiltshire equations model what happens when two orbiting bodies are close to each other. Typically, one spacecraft is fixed as the primary craft, which is called the target, and the secondary spacecraft is called the chaser. The Clohessey-Wiltshire equations model the movement of the chaser relative to the target. In other words, in this frame of reference, the target's position is always considered to be (0,0, 0). The functions = (}, {t), and = (t) give the coordinates of the chaser's position at time . Likewise, their derivatives and second derivatives give the velocity and acceleration of the chaser relative to the velocity and acceleration of the target. This is useful because the movement of the chaser may be significant relative to the position of the target, even if the movement is small and insignificant relative to the body which it is orbiting. The system of equations is as follows (where all primes denote derivatives with respect to time, #): " 2wy 3wtz =10 Yy 4+ 2wx' =0 2wl =0 In the previous equations, w is a constant which denotes the mean angular velocity of the target spacecraft. It can be calculated by w = 2?" where T is the orbital period of the target body. Note that the last of the three equations is a differential equation in only z(f} and can therefore be solved as a stand-alone differential equation. On the other hand, the first two equations contain derivatives of x as well as derivatives of 1, so they form a system of differential equations. In Medule 7, we will focus on solving the third differential equation to find an explicit expression for z{t} in order to get one compenent of the position of the chaser. Then, in Module 8, we will learn how to use the Laplace transform to solve a system of differential equations. We will use this method to solve the system of equations determined by the first two equations simultanecusly to find z(t) and y(t) so that by the end of Madule 8, we will have explicit equations of motion for the chaser. The specific details of your situation will vary from student to student and will be provided by your instructor. Your instructor will provide each of you with the following: 1. The orbital period of the spacecraft 2. Avector (2(0), y(0), 2(0)) representing the initial position of the chaser relative to the target 3. Avector (' (0}, %' (0), 2' (0)) representing the initial velocity of the chaser relative to the target This information will create an initial value problem that you can solve ta find the relative position of the spacecraft. You will be using the Laplace transform to solve these equations. In your work, you may refer to the Laplace transform of z(t) as Z{s), the transform of z(t) as X(#) and the transform of y(t) as '(s). Make sure to be careful with your work so that you use capital letters in the correct places. As you calculate, you may choose to use exact values or round to 4 significant figures. Also, note that the work in Module B can get a little messy, so just make sure to go slowly and be careful with your work. Use the values that your instructor has provided you to solve the initial value problem defined by the third equation (2 + wz = 0) to find a function that gives the z component of the position of the chaser relative to the target. Make sure to state the values that you were given. While this IVP can be solved using other methods, for the sake of this discussion, you must use the Laplace transform method to solve the differential equation. Submit your initial post to the Module 7 & 8 - Assignment: The Clohessey-Wiltshire Equations to have your answer run through Turnitin and post it to the discussion forum. Your initial post should be 100 to 300 words. Please proceed to the By Day 7 of Module 7 section. Use the values which you were given by your instructor at the beginning of Module 7 to take the Laplace transform of both of the first two equations and get an algebraic system of equations. Solve this system of equations so that you have one algebraic equation containing only X7(s), but not (s), and a second one which contains just '{s) but not X{s). It is recommended that you post this step for feedback before moving on to using the inverse Laplace to recover x(t) and y(t) because the algebra can be messy, and this is a common place for errors. Your second post should be 100 to 200 words. Please proceed to the By Day 7 of Module 8 section. By the end of Module 8. you must post both the following: Aresponse to one of your peers with observations that add to a meaningful conversation about the discussion topic. The conversation may include getting help from classmates or providing feedback on their solutions, or comparing methods. = Your response should be 100 to 250 words. = Your final position functions (z(t}, y(t), 2(t)). You must explain any work which was not previously explained in your previous posts. If you have received feedback over the course of Modules 7-8, you may need to revise some of your previous work before presenting a final sclution. You must also include an explanation of what these position equations mean in terms of the two spacecraft. Will they come close? Will they collide? = Your final post should be 100 to 250 words