pset_II-06-2

# pset_II-06-2 - result Which one contributes the least 3 A...

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MMAE 350 D. Rempfer Problem Set II Page 1 of 1 Due Date: September 20, 2006 1. Write a program (in a computer language of your choice, or even on your pocket calculator) that calculates the machine epsilon of your computer. What do you ﬁnd for ε ? 2. If one neglects the e ect of surface tension, then the frequency of the oscillations of air bubbles in a liquid (which cause, e. g., the sounds made by water ﬂowing over the rocks in small streams) is approximately given by ω 0 = 1 R 0 ± 3 γp 0 ρ ² 1 2 , where R 0 is the bubble radius, γ is the ratio of speciﬁc heats of the gas, p 0 is the hydrostatic pressure at the location of the bubble, and ρ is the density of the surrounding liquid. We want to get a rough idea of what frequencies to expect for typical bubbles in a brook. The data that we can assume for this are: R 0 = 0 . 01 ± 0 . 005 m, γ = 1 . 4, ρ = 1000 ± 100 kg/m 3 , p 0 = 103 , 000 ± 300 N/m 2 . a. What is the average frequency you would expect from these data? b. Using a linear error analysis, what are the upper and lower bounds for the frequency? c. Which of the above data uncertainties contributes the most to the relative uncertainty in the
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Unformatted text preview: result? Which one contributes the least? 3. A missile leaves the ground with an initial velocity v forming an angle ϕ with the vertical. The maximum desired altitude is h = αR , where R is the radius of the earth. The laws of mechanics can be used to show that sin( ϕ ) = (1 + α ) s 1-α 1 + α ± v e v ² 2 , where v e is the escape velocity of the missile. It is desired to ﬁre the missile and reach the design maximum altitude with an accuracy of ± 2%. Determine the admissible range of values for ϕ , both using ﬁrst-order (linear) error analysis, and exactly, if v e /v = 2 and α = . 25. 4. Give the condition numbers κ for the following functions: a. f ( x ) = 1 p | x + 1 | -1 , for x = . 0001 , b. f ( x ) = e-x , for x = 9 , c. f ( x ) = p x 2 + 1-x, for x = 200 , d. f ( x ) = e x-1 x , for x = . 01 , e. f ( x ) = sin( x ) 1 + cos( x ) , for x = 1 . 001 π....
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