Escape Velocity Eulers Method: In of Section 7-4, you learned that the acceleration

Chapter 7, Problem 9

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Escape Velocity Eulers Method: In of Section 7-4, you learned that the acceleration (rate of change of velocity) of a spaceship moving away from Earth decreases due to Earths gravity according to the differential equation where v is the spaceships velocity, in km/s, and r is the distance from Earths center, in earth-radii. The slope field for this differential equation is reproduced here (Figure 7-5i) for your use. Figure 7-5i a. Starting with an initial velocity of 12 km/s at Earths surface (r = 1), use your Eulers method program with dr = 0.6 to show that the velocity appears to become negative, meaning that the spacecraft will stop and start to fall back to Earth. b. Starting at (1, 12) as in part a, use dr = 0.1 to show that the velocity appears to be positive when r = 20 and the values of v seem to be leveling off. c. Solve the differential equation algebraically and use the solution to find the actual velocity at r = 20. Explain why Eulers method using a relative large value of dr (as in part a) erroneously predicts that the spaceship will not escape Earths gravity. d. Show that if the spaceship starts from Earths surface with an initial velocity less than the escape velocity, ve = km/s,then the spacecraft will eventually reversedirections and start to fall back towardEarth. Show also that if the initial velocity isgreater than ve , then the velocity approachesa positive limit as r approaches infinity andthe spacecraft never returns. What is theescape velocity in miles per hour?