nagle_differential_equations_ISM_Part21

nagle_differential_equations_ISM_Part21 - Chapter 3 Setting...

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Unformatted text preview: Chapter 3 Setting v = 5 when t = 0, we find that 5 = 10+ C so that C =- 5 and v ( t ) = 10- 5 e- 6 t/ 5 . The limiting velocity of the sailboat under these conditions is lim t →∞ v ( t ) = lim t →∞ ( 10- 5 e- 6 t/ 5 ) = 10 (m / sec) . To find the equation of motion, we integrate v ( t ) using the initial condition x (0) = 0. x ( t ) = t Z ( 10- 5 e- 6 s/ 5 ) ds = 10 s + 25 6 e- 6 s/ 5 t = 10 t + 25 6 ( e- 6 t/ 5- 1 ) . 24. Dividing the equation by m- αt yields dv dt =- g + αβ m- αt ⇒ v ( t ) = t Z- g + αβ m- αs ds + v (0) = [- gs- β ln( m- αs )] | s = t s =0 =- gt + β ln m m- αt , where we used the condition 0 ≤ t < m /α so that m- αt > 0. Since the height h ( t ) of the rocket satisfies h (0) = 0, we find h ( t ) = t Z v ( s ) ds = t Z- gs + β ln m m- αs ds =- gs 2 2 + βs ln m + β α ( m- αs ) ln m- αs e s = t s =0 = βt- gt 2 2- β α ( m- αt ) ln m m- αt . EXERCISES 3.5: Electrical Circuits 2. Capacitor voltage =- 10000 100000001 cos 100 t + 100000000 100000001 sin 100...
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This note was uploaded on 02/12/2011 for the course MA 221 taught by Professor Mazmani during the Spring '08 term at Stevens.

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nagle_differential_equations_ISM_Part21 - Chapter 3 Setting...

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