Explanation Given: Weight of the skydiver is 125 pounds , altitude is 15 , 000 feet and the value of k is 0.5 . Calculation: Consider a differential equation that describes a falling object with air resistance proportional to the instantaneous velocity. m d v d t = m g − k v d v d t + k m v = g (1) Here, g is the acceleration due to gravity, k is proportionality constant and m is mass. Compare the Equation (1) with d v d t + p ( t ) v = Q ( t ) . The value of p ( t ) is k m and Q ( t ) is g . Calculate the integrating factor. I F = e ∫ p ( t ) d t = e ∫ k m d t = e k m t The solution of the differential equation is, v e k m t = ∫ g e k m t d t v e k m t = g m k e k m t + c v = g m k + c e − k m t (2) Find the anti derivative of Equation (2) to find the position. y = ∫ ( g m k + c e − k m t ) d t y = g m k t − m k c e − k m t + c 1 (3) Calculate the total mass of the sky diver. Mass = weight g = ( 125+35 ) 32 = 160 32 = 5 Substitute 5 pounds for m and 0.5 for k in Equation (2). v f = ( 32 ) ( 5 ) 0.5 + c e − 0.5 5 t = 320 + c e − 0.1 t (4) Here, v f is free fall velocity. Substitute 0 for t and v f ( 0 ) = 0 in Equation (4). 0 = 320 + c e − 0.1 ( 0 ) − 320 = c Substitute − 320 for c in Equation (4). v f = 320 − 320 e − 0.1 t = 320 ( 1 − e − 0.1 t ) (5) Substitute 15 for t in Equation (5). v f = 320 ( 1 − e − 0.1 ( 15 ) ) = 320 ( 0.77 ) = 248.59 Thus, the velocity of the skydiver at t = 15 sec is 248.59 ft / sec . Calculate the free fall position of the skydiver. Substitute 5 for m , 0.5 for k and 32 ft / sec 2 for g in Equation (3). y f = ( 32 ) ( 5 ) 0.5 t − 5 0.5 ( − 320 ) e − 0.5 5 t + c 1 y f = 320 t + 3200 e − 0.1 t + c 1 . (6) Here, y f is the freefall position. Substitute 0 for t and y f ( 0 ) = 0 in Equation (6). 0 = 320 ( 0 ) + 3200 e − 0.1 ( 0 ) + c 1 c 1 = − 3200 Substitute − 3200 for c 1 in Equation (6). y f = 320 t + 3200 e − 0.1 t − 3200 (7) Substitute 15 for t in Equation (7)

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ISBN: 9781284105902

Author: Dennis G. Zill

Publisher: Jones & Bartlett Learning

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Chapter 2.7, Problem 38E

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Calculate the velocity and the distance travelled after 20 sec and the total time taken by the skydiver to reach the ground.

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Chapter 2.7, Problem 37E

Chapter 2.7, Problem 39E

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Chapter 2 Solutions

Advanced Engineering Mathematics

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