Explanation Given: The average velocity of falling raindrop is 7 miles / hour . The specific weight of a raindrop is 62.4 lbs / ft 3 . The volume of raindrop is 0.000000155 ft 3 . The falling object with air resistance is proportional to its instantaneous velocity. The differential equation is m d v d t = m g − k v . Here, k is the constant of proportionality for air resistance. The general solution of the above equation is v ( t ) = m g k + ( v 0 − m g k ) e − k t m . Here, v 0 is the initial velocity of the falling object. If the falling object starts at rest like a raindrop, then v 0 = 0 and the solution becomes, v ( t ) = m g k ( 1 − e − k t m ) … … ( 1 ) The falling object with air resistance is proportional to square of its instantaneous velocity. The differential equation is m d v d t = m g − k v 2 . Here, k is the constant of proportionality for air resistance. The general solution of the above equation is v ( t ) = m g k tanh ( k g m t + tanh − 1 ( k g m v 0 ) ) . Here, v 0 is the initial velocity of the falling object. If the falling object starts at rest like a raindrop, then v 0 = 0 and the solution becomes v ( t ) = m g k tanh ( k g m t ) … ( 2 ) . The specific gravity is written as, ρ = m g V . Substitute 62.4 lbs / ft 3 for ρ , 32 for g and 1.55 × 10 − 7 for V in above equation. 62.4 = 32 m 1.55 × 10 − 7 m = 1.55 × 10 − 7 × 62.4 32 = 3.0225 × 10 − 7 Substitute 3.0225 × 10 − 7 for m and 32 for g in equation (1). v ( t ) = 3.0225 × 10 − 7 × 32 k ( 1 − e − k t 3.0225 × 10 − 7 ) = 9.672 × 10 − 6 k ( 1 − e − k t 3.0225 × 10 − 7 ) … … ( 3 ) Substitute 3

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

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Differential Equations with Boundary-Value Problems (MindTap Course List)

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To check: Whether the average velocity of rain drop is consistent with the models or not.

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