Explanation Procedure used: To find the acceleration of an object whose coordinates are given in parametric form: 1 Determine the x – component of acceleration a x by finding the derivative d v x d t where v x is the x – component of velocity. 2. Determine the y – component of acceleration a y by finding the derivative d v y d t where v y is the y – component of velocity. 3. Compute the magnitude of acceleration by using the formula, a = a x 2 + a y 2 and the direction of acceleration by the formula, tan θ = a y a x . Calculation: Given that the equations x = 10 ( 1 + t 4 − 1 ) and y = 40 t 3 2 ( x and y in meters , t in seconds ) , which describes the path of a spacecraft for the first 100s after launch. Obtain the x – component of velocity by computing the derivative, d x d t as shown below. v x = d x d t = d d t ( 10 ( 1 + t 4 − 1 ) ) = d d t ( 10 ( 1 + t 4 ) 1 2 − 10 ) = 10 ( 1 2 ) ( 1 + t 4 ) − 1 2 ( 4 t 3 ) ( Since d u n d x = n u n − 1 d u d x ) = 20 t 3 ( 1 + t 4 ) 1 2 Now obtain the x – component of acceleration by computing the derivative, d v x d t as shown below. a x = d v x d t = d d t ( 20 t 3 ( 1 + t 4 ) 1 2 ) = ( 1 + t 4 ) 1 2 20 ( 3 t 2 ) − 20 t 3 ( 1 2 ) ( 1 + t 4 ) − 1 2 ( 4 t 3 ) ( ( 1 + t 4 ) 1 2 ) 2 = ( 1 + t 4 ) 1 2 60 t 2 − 40 t 6 ( 1 + t 4 ) − 1 2 ( 1 + t 4 ) Obtain the y – component of velocity by computing the derivative, d y d t as shown below. v y = d y d t = d d t ( 40 t 3 2 ) = ( 3 2 ) 40 t 3 2 − 1 ( Since d u n d x = n u n − 1 d u d x ) = 120 2 t 1 2 = 60 t 1 2 Now obtain the y – component of acceleration by computing the derivative, d v y d t as shown below. a y = d v y d t = d d t ( 60 t 1 2 ) = 60 ( 1 2 ) t − 1 2 ( Since d u n d x = n u n − 1 d u d x ) = 30 t − 1 2 When t = 10 s , the x – component of acceleration becomes, a x | t = 10 = ( 1 + ( 10 ) 4 ) 1 2 60 ( 10 ) 2 − 40 ( 10 ) 6 ( 1 + ( 10 ) 4 ) − 1 2 ( 1 + ( 10 ) 4 ) = ( 10001 ) 1 2 ( 6000 ) − ( 40000000 ) ( 10001 ) − 1 2 10001 ≈ 200049

11th Edition

ISBN: 9780134437705

Author: Washington

Publisher: PEARSON

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Chapter 24.3, Problem 23E

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The resultant acceleration of the spacecraft.

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Chapter 24.3, Problem 22E

Chapter 24.3, Problem 24E

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Chapter 24 Solutions

Basic Technical Mathematics

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