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quiz_11A - cv Solve this dierential equation to nd the...

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Math 213 — Quiz 11 Name Solution 1. An object with mass m is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If s ( t ) is the distance dropped after t seconds, then the speed is v ( t ) = s 0 ( t ) and the acceleration is a ( t ) = v 0 ( t ). If g is the acceleration due to gravity, then the downward force on the object is mg - cv , where c is a positive constant. Newton’s Second law gives m dv dt = mg -
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Unformatted text preview: cv. Solve this dierential equation to nd the velocity at time t . This is a linear dierential equation when written in the form dv dt + c m v = g. The integration factor is I ( t ) = exp( Z c m dt ) = e ct/m . Thus the general solution is v ( t ) = 1 I ( t ) Z I ( t ) g dt = e-ct/m gm c e ct/m + K = gm c + Ke-ct/m where K is an arbitrary constant....
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