By K. D. Stroyan
Calculus utilizing Mathematica
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Additional info for Calculus Using Mathematica. Scientific Projects and Mathematical Background
1: Free Fall with Air Friction Light and heavy objects fall at the same speeds if we can neglect air friction. Often we can't neglect that. ma NoteBook of Chapter 10 of the main text. We know that a parachute radically affects the speed 61 with which an object falls - air friction often does matter. ma is shown above. Notice that it 'flattens out' as time increases. 2. ma NoteBook from t = 0 to t = \, from t=\ to t = 1, etc. What happens to speed as time increases? 2) Use your velocity list to compute the accelerations a(t) from t = \ to t = 1, form t = 1 to t = | , etc.
Suppose t is fairly large, so that the fast exponential is negligible, b\e~hlt « 0. What is the graph o/Log[cß[t]] « \jOg\b2e~h2t] for an interval oft values in this range? (Hint: Look at the previous figure and justify the graphically obvious feature of the tail of the plot. ) Show that the slope of the near-linear tail of the Log[cß[t]] graph is approximately —h2, in fact, the line has the form Log[cß[i]] « ß2 — h2t for a constant ß2, when t^>0 How could you find b2 by extending the linear tail back to the CB axis?
3) Light travels at speed c, so the time required for light to go from (a, b) to (x, 0) is the distance over c. Similarly, the time required to go from (x, 0) to (w, v) is that distance over c.