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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 6.055J/2.038J (Spring 2008) Solution set 4 Do the following warmups and problems. Due in class on Friday, 04 Apr 2008 . Open universe: Collaboration, notes, and other sources of information are encouraged. However, avoid looking up answers until you solve the problem (or have tried hard). That policy helps you learn the most from the problems. Bring a photocopy to class on the due date , trade it for a solution set, and figure out or ask me about any confusing points. Your work will be graded lightly: P (made a reasonable effort), D (did not make a reasonable effort), or F (did not turn in). Warmups 1. Minimum power In lecture we estimated the ight speed that minimizes energy consumption. Call that speed v E . We could also have estimated v P , the speed that minimizes power consumption. What is the ratio v P / v E ? The zillions of constants (such as ) clutter the analysis without changing the result. So Ill simplify the problem by using a system of units where all the constants are 1. Then the energy is 1 E v 2 v 2 , + where the first term is from drag and the second term is from lift. The power is energy per time, and time is inversely proportional to v , so P Ev and 1 + . P v 3 v The first term is the steep v 3 dependence of drag power on velocity (which we used to estimate the world-record cycling and swimming speeds). The energy expression is unchanged when v 1 / v , so it has a minimum at v E = 1. To minimize the power, use calculus (ask me if you are curious about calculus-free ways to minimize it): dP 1 dv 3 v 2 v 2 = , therefore v P = 3 1 / 4 (roughly 3/4), which is also the ratio v P / v E . So the minimum-power speed is about 25% less than the minimum-energy speed. That result makes sense. Drag power grows very fast as v increases much faster than lift power decreases so its worth reducing the speed a little to reduce the drag a lot. If you dont believe the simplification that I used of setting all constants to 1 and it is not imme- diately obvious that it should work then try using this general form: B E Av 2 + v 2 , where A and B are constants. Youll find that v E and v P get the same function of A and B , which disappears from the ratio v P / v E . 2 Solution set 4 / 6.055J/2.038J: Art of approximation in science and engineering (Spring 2008) 2. Solitaire You start with the numbers 3, 4, and 5. At each move, you choose any two of the three numbers call the choices a and b and replace them with 0 . 8 a . 6 b and 0 . 6 a + . 8 b . The goal is to reach 4, 4, 4. Can you do it? If yes, give a move sequence; if no, show that you cannot....
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sol04 - MIT OpenCourseWare http://ocw.mit.edu 6.055J /...

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