Explanation Given data: Initial downward velocity is 10 m s , initial displacement is 500 m. Consider the downward acceleration as, a = { 9 − 0.9 t if 0 ≤ t ≤ 10 0 if t > 10 Formula Used: Write the expression for acceleration function a ( t ) . a ( t ) = v ′ ( t ) (1) Here, v ′ ( t ) is first derivative of velocity function v ( t ) . Write the expression for velocity function v ( t ) . v ( t ) = s ′ ( t ) (2) Here, s ′ ( t ) is first derivative of displacement function s ( t ) . Antiderivative of t n is t n + 1 n + 1 , 0 is K , and 1 is t . Calculation: Consider upward direction is indicated by positive sign and downward direction by negative. The initial velocity of raindrop is 10 m s downwards, that is v ( 0 ) = − 10 and consider the initial displacement as 500 m, that is s ( 0 ) = 500 . Write the expression for acceleration ( a 1 ( t ) ) considering time as 0 ≤ t ≤ 10 . a 1 ( t ) = − ( 9 − 0.9 t ) = 0.9 t − 9 Substitute v ′ 1 ( t ) for v ′ ( t ) and 0.9 t − 9 for a ( t ) in equation (1), v ′ 1 ( t ) = 0.9 t − 9 Antiderivate the expression with respect to t . v 1 ( t ) = − 9 t + 0.9 ( 1 2 t 2 ) + C v 1 ( t ) = − 9 t + 0.45 t 2 + C (3) Here, C is arbitrary constant. Substitute 0 for t in equations (3), v 1 ( 0 ) = − 9 ( 0 ) + 0.45 ( 0 ) 2 + C = 0 + C = C Substitute –10 for v ( 0 ) , C = − 10 Substitute –10 for C in equation (3), v 1 ( t ) = − 9 t + 0.45 t 2 − 10 (4) Substitute s ′ 1 ( t ) for s ′ ( t ) and − 9 t + 0

Essential Calculus: Early Transcendentals

2nd Edition

ISBN: 9781133112280

Author: James Stewart

Publisher: Cengage Learning

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Chapter 4.7, Problem 49E

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The time taken by raindrop to fall.

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The time taken by raindrop to fall.

Solution Summary: The author calculates the time taken by raindrop to fall from 500 m above the ground.

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The time taken by raindrop to fall.

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