differential equations. Complete the Landing Vehicle project shown on the next page. Your submission must go beyond answering the six questions, and should take the form of a technical report comprising 4 parts: listed in introduction

Part 1: Background and Overview: Describe the problem that NASA is facing, and include any necessary background material. This section sets the stage for the rest of the report and explains the main research question being addressed. It should be two to three paragraphs long. Part 2: Analysis Approach: Describe how you intend to analyze the situation to properly advise NASA. This section should include a description of the modeling techniques used, including any assumptions you made. Your answer to question #1 from the next page should go in this section. You should also describe the solution techniques you will use to solve the differential equation in this section. In a more complex project, this section could be quite lengthy, but for this project I expect it 1 page Part 3: Analysis: This is where you put your analysis. This section will be the longest section and should include equations and graphs as necessary. This is where all of the work you do to answer questions #2-5 should go. Part 4: Conclusions: This section should be used to interpret the work you did in Part 3. This is where you answer the main research question you asked in Part 1. You should answer the questions about whether or not the lander will bottom out on the moon and Mars in this section. You should also answer question #6 in this section. Make any recommendations for NASA that result from your analysis. This section of the report should be about three to five paragraphs long, although it could be longer if you have more recommendations. Landing Vehicle NASA is planning a mission to Mars. To save money, engineers have decided to adapt one of the moon landing vehicles for the new mission. However, they are concerned about how the different gravitational forces will affect the suspension system that cushions the craft when it touches down. The acceleration resulting from gravity on the moon is 1.6 m/sec2 , whereas on Mars it is 3.7 m/sec2 . The suspension system on the craft can be modeled as a damped spring-mass system. In this case, the spring is below the moon lander, so the spring is slightly compressed at equilibrium, as shown in Figure 7.12. Figure 7.12 The landing craft suspension can be represented as a damped spring-mass system. (credit “lander”: NASA) We retain the convention that down is positive. Despite the new orientation, an examination of the forces affecting the lander shows that the same differential equation can be used to model the position of the landing craft relative to equilibrium: mx″ + bx′ + kx = 0, where m is the mass of the lander, b is the damping coefficient, and k is the spring constant. 1. The lander has a mass of 15,000 kg and the spring is 2 m long when uncompressed. The lander is designed to compress the spring 0.5 m to reach the equilibrium position under lunar gravity. The dashpot imparts a damping force equal to 48,000 times the instantaneous velocity of the lander. Set up the differential equation that models the motion of the lander when the craft lands on the moon. 2. Let time t = 0 denote the instant the lander touches down. The rate of descent of the lander can be controlled by the crew, so that it is descending at a rate of 2 m/sec when it touches down. Find the equation of motion of the lander on the moon. 3. If the lander is traveling too fast when it touches down, it could fully compress the spring and “bottom out.” Bottoming out could damage the landing craft and must be avoided at all costs. Graph the equation of motion found in part 2. If the spring is 0.5 m long when fully compressed, will the lander be in danger of bottoming out? 4. Assuming NASA engineers make no adjustments to the spring or the damper, how far does the lander compress the spring to reach the equilibrium position under Martian gravity? 5. If the lander crew uses the same procedures on Mars as on the moon, and keeps the rate of descent to 2 m/sec, will the lander bottom out when it lands on Mars? 6. What adjustments, if any, should the NASA engineers make to use the lander safely on Mars?