1CHAPTER 11.1We will illustrate two different methods for solving this problem: (1) separation of variables, and (2)Laplace transform.dvcgvdtmSeparation of variables:Separation of variables gives1dvdtcgvmThe integrals can be evaluated asln/cgvmtCcmwhereC= a constant of integration, which can be evaluated by applying the initial condition to yieldln(0)/cgvmCcm which can be substituted back into the solutionlnln(0)//ccgvgvmmtcmcmThis result can be rearranged algebraically to solve forv,( /)( /)(0)1c m tc m tmgvveecwhere the first part is the general solution and the second part is the particular solution for the constantforcing function due to gravity. For the case where,v(0) = 0, the solution reduces to Eq. (1.10)( /)1c m tmgvecLaplace transform solution:An alternative solution is provided by applying Laplace transform to thedifferential equation to give( )(0)( )gcsV svV ssmSolve algebraically for the transformed velocityPreview Mode
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