E- Calculus & Anyl Geometry II

In this assignment, you will prepare and submit a document with your responses to this activity. Your solution should not only be an answer to the question but should include a convincing argument for the correctness of your proposed solution. Any claims you make should be backed up by evidence, and you should include a justification that links the evidence to the claim and incorporates the appropriate physical interpretations of Riemann Sum approximations. For this assessment, you only need to provide evidence and justification for your answer to Question 6, however you should fully answer Questions 1-5. The evidence and justification to your Question 6 answer should incorporate your answers to Questions 1-5. Your evidence should include a desсrіption of the information gained from the applet from which you determined your answer. You must include the calculations by which you determine relevant parameters (such as the gravitational constant) and also any calculations by which you determine the values of relevant integrals and Riemann Sums. You must also confirm any Riemann Sum or integral calculation using MATLAB, or using the applet. Your justification should specifically refer to the different geometric or physical interpretations possible from Riemann Summation. You should also label any relevant definite integrals and Riemann Sums in detail, being sure to provide contextual interpretations for the symbols. You should also describe why the provided Riemann Sums and definite integrals calculate the desired energies or areas. In this assessment, we are estimating the energy requirements on the process of sending a 1 kg payload from the surface of the Earth to a satellite in orbit around the planet. Context: Gravitational forces depend on the relative masses of objects as well as the distances separating the centers of the objects. The farther two objects are from each other, the easier it is to move the objects apart from each other. While it technically requires more energy to lift an object 1 meter into the air while at sea level than it does to perform the same action while standing atop a skyscraper, the altitude difference between these places is so small compared to the radius length of the Earth that the difference is negligible. Moving an object into space, however, will result in significant differences to the expended energy because of the breadth of distance over which an object is moved away from the planet. Newton postulated that the force of gravity on a given mass is proportional to the reciprocal of the square distance between that mass and the center of the earth. For each object, there is a constant k such that at a distance D from Earth′s center, the gravitational force on that object is F(D)＝kD2. Recall that when a force F is uniformly applied to move an object across some distance D, the energy required to move the object is dE＝F⋅dD. GeoGebra Application Instructions: Simulated in this applet are approximations of the object′s flight to the satellite. Displayed on the right screen is the curve of the gravitational force for the particular object being sent to the satellite. The given rectangles visually represent the associated Riemann Sum approximations of the area underneath the force curve. Displayed on the left screen is the object′s path from Florida to the satellite. You may alter the ″Left″, ″Mid″, and ″Right″ slider to create a left, midpoint, or right Riemann Sum approximation. You may alter the ″Number of Segments″ slider to select the number of intervals between the satellite and the Earth′s surface. You may alter the ″Segment to Evaluate″ slider to view a specific interval of distance for the object′s path on the left and the corresponding rectangle on the right. Task Your overall goal is to calculate the energy required to send a 1 kg object from the surface of Earth to the satellite located 42.16×106 meters from the center of Earth. You must answer the following questions, with Question 6 asking you to make the energy calculation. Answering Questions 1-5 is not only required, but each will also help you achieve the ultimate goal of calculating the exact energy. Note: Assume the location from which the object is being launched is 6.37×106 meters from the center of the earth. Question 1) Use the applet to calculate the third term in R8. What does the depicted area represent in the context of sending the object to orbit? What does the height of the rectangle represent in this context? What does the width of the rectangle represent in this context? Question 2) Calculate and interpret L5. How far is the object traveling within each interval of distance for this approximation? Question 3) Approximately how much force is being uniformly applied in the approximation given by the 7th term in M12? Does the energy given in this approximation overestimate or underestimate the true energy required to move the object over this interval of distance? Why? Question 4) How accurate are the approximations in L3 and R3? Note that these sums either overestimate or underestimate the true energy requirement, and so you need to calculate both to measure the possible error between these approximations and the true energy. Question 5) Use the object′s weight at the surface of the earth (assume gravitational acceleration here to be 9.8ms2) to find Newton′s proportional constant k for this object. Question 6) Write a definite integral that calculates the energy required to send the object to orbit, then use the Fundamental Theorem of Calculus to calculate the energy. Label each symbol in the integral acco ending a 1 kg payload from the surface of the Earth to a satellite in orbit around the prding to its contextual meaning, and show your use of the Fundamental Theorem of Calculus. You must confirm this calculation with MATLAB. Question 7) Calculate the fraction of the object′s total distance traveled at which half of the total required energy is expended. You must confirm this calculation with MATLAB.