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Problem set 1Ge 108October 2, 20081 Fun with expone ntialsVerify equation (1.8), thatddx(1/2)x= ln(1/2)(1/2)x2 Random DEsSolve the following differential equations by using a n integrating factor:(a) xdydx+ (1 − x)y = xex(b) tdxdt− kx = t2(k is a constant)For (b), what if k = 2?3 Hydrostatic equilibriumIn the troposphere (first ∼ 10 km of the earth’s atmosphere), the atmospherictemperature decreases by 6.5 K/km. Assuming a surface temperature of T0=300K and plugging this function for T (z) into the equation of hydrostaticequilibrium, what is the atmospheric pressure on top of Mt. Wilson (elevation∼ 2000 m) when the pressure at sea level is 105Pascal? How much less denseis the atmosphere there than at sea level? What is the scale height at sealevel? What is the pressure one scale-height above sea level?1For this problem, you will need t o know that the mean molecular mass ofair (which is mostly N2) is 29, and that, in the appropriate units, R = 8 310J/kg/K4 Numeric solutionsOn the Ge 108 web page (www.gps.caltech.edu/∼mbrown/classes/ge108)click on “problem set # 1” a nd copy the program “numeric.m” to yourcomputer account.This program does a numeric solution of the 2- element radioactive decayproblem that we solved in class and also computes the exact solution. Runthe program for tm= 1000 yrs, tn= 10 0 yrs, and ∆t=1 yr. How accurate isthe numeric solution compared to the exact solution?Now try ∆t = 10, ∆t = 50 yrs, and ∆t = 100 yrs. How accurate are thesolutions now?Go back to the case for ∆t = 1 yr, but now change tmto 100 yrs. and tnto 10 yrs. What a ccuracy do you get?For this equation, at least, can you come up with a general principle forhow small ∆t needs to be to get reasonable accuracy for your solution?5 Making up differential equ ations(a) Assume that for a spherical drop of water, t he evaporation rate is pro-portional to the surface area of the drop. Write a differential equation forthe radius of the drop as a function of time. What are the units of theproportionality constant that you had to use? Solve the equation!(b) A lake has a volume of 106m3and a surface area of 6 × 104m2. Waterflows into the lake at a n average rate o f 0.005 m3/s. The amount of waterthat evaporates yearly from the lake is equivalent in volume to the lake’s topmeter of water. The lake is a lready full, so it can get no deeper than itscurrent depth. Any additional input of water causes lake water to spill overa dam. Initially, the la kewater is pristine, but at a certain time a soluable,noncodistilling (jargon for “ it doesn’t evaporate, but it does flow away ifthe water flows away”. Think of, for example, salt) is discharged into thelake at a steady rate of 40 tons/year (1 ton = 103kg). Derive a formulafor the concentration of pollutant in the lake a s a function of time. How2much pollutant will the lake contain as time approaches infinity? Plot thisfunction, using t he real values in


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CALTECH GE 108 - Problem set 1

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