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Chapter2Section3 - 2.3 Acceleration-Velocity Models...

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2.3: Acceleration-Velocity Models Vertical Motion without Air Resistance m F G Recall: F = m a = m d v dt . So, with y H 0 L = y 0 and v H 0 L = v 0 , we have Force of Gravity is F G = - m g m d v dt = F G = - m g d v dt = - g . This has solutions v H t L = - g t + C 1 v H t L = - g t + v 0 = d y dt y H t L = - g 2 t 2 + v 0 t + C 2 y H t L = - g 2 t 2 + v 0 t + x 0 .
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ª Example Assume that v 0 = 0, y 0 = 20. vNoAir = v @ t D .DSolve @8 v' @ t D - 9.8, v @ 0 D 0 < , v @ t D , t D 8 - 9.8t < yNoAir = y @ t D .DSolve @8 y' @ t D vNoAir @@ 1 DD , y @ 0 D 20 < , y @ t D , t D 9 20 - 4.9t 2 = sol1 = t .Solve @ yNoAir @@ 1 DD 0, t D 8 - 2.02031, 2.02031 < 2 Chapter2Section3.nb
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Plot @8 vNoAir, yNoAir < , 8 t, 0, sol1 @@ 2 DD< , PlotStyle fi 88 Blue, Thick, Dashed < , 8 Blue, Thick << , AxesLabel fi 8 t, 8 y, v << , PlotLabel fi "Blue = No Air resistance" D 0.5 1.0 1.5 2.0 t - 20 - 10 10 20 8 y , v < Blue = No Air resistance Chapter2Section3.nb 3
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Vertical Motion with Air Resistance (proportional to velocity) m F G F R Motion is DOWN m F G F R Motion is UP Recall: F = m a = m d v dt . Force of Gravity is F G = - m g Force of Air Resistance is F R = - k v where k is a constant chosen so that the force, F R , always opposes motion. (So, in the first picture, k < 0 while in the second picture, k > 0.) The Net Force if the particle is falling! F = F G + F R = - m g - k v So, with y H 0 L = y 0 and v H 0 L = v 0 , we have m d v dt = F = - m g - k v d v dt = - g - Ρ v where Ρ = k m and which has solution v H t L = K v 0 + g Ρ O e t - g Ρ Notice that lim t z ¥ v H t L = - g Ρ = v Τ . Thus, the two solutions can be written as v H t L = K v 0 + g Ρ O e t - g Ρ = H v 0 - v Τ L e t + v Τ y H t L = y 0 + v Τ t + 1 Ρ H v 0 - v Τ L I 1 - e t M . Terminal Speed = v Τ = g Ρ = m g k . 4 Chapter2Section3.nb
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ª Example Assume that v 0 = 0, y 0 = 20, Ρ = 1. vAir = v @ t D .DSolve @8 v' @ t D == - v @ t D - 9.8, v @ 0 D == 0 < , v @ t D , t D 9 ª - 1.t I 9.8 - 9.8 ª 1.t M= yAir = y @ t D .DSolve @8 y' @ t D vAir @@ 1 DD , y @ 0 D 20 < , y @ t D , t D 9 ª - 1.t I - 9.8 + 29.8 ª 1.t - 9.8 ª 1.t t M=
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