Explanation Given: The acceleration vector a ( t ) = 〈 t , e − t , 1 〉 , initial velocity and initial position are 〈 u 0 , v 0 , w 0 〉 = 〈 0 , 0 , 1 〉 and 〈 x 0 , y 0 , z 0 〉 = 〈 4 , 0 , 0 〉 respectively. Definition used: Let the position vector of an object moving in three dimensional space be given by r ( t ) = 〈 x ( t ) , y ( t ) , z ( t ) 〉 for t ≥ 0 . Then, thevelocity of the object v ( t ) is the derivative of the position vector, r ′ ( t ) = 〈 x ′ ( t ) , y ′ ( t ) , z ′ ( t ) 〉 . The acceleration of the objects is the derivative of the velocity of the object a ( t ) = v ′ ( t ) . Then, velocity can be obtained from antiderivative of acceleration vector as v ( t ) = ∫ ( a ( t ) ) d t and the position vector can be obtained from antiderivative of the velocity vector as r ( t ) = ∫ v ( t ) d t . Calculation: Compute the antiderivative of the acceleration vector a ( t ) = 〈 t , e − t , 1 〉 to obtain the velocity vector of the motion, v ( t ) = ∫ ( a ( t ) ) d t = ∫ 〈 t , e − t , 1 〉 d t = 〈 t 2 2 , e − t − 1 , t 〉 + C = 〈 t 2 2 , − e − t , t 〉 + C Here, C is the initial velocity of 〈 u 0 , v 0 , w 0 〉

3rd Edition

ISBN: 9780134856971

Author: Briggs

Publisher: PEARSON

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Chapter 14.3, Problem 50E

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To find: The velocity vector and position vector, for t≥0 .

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Chapter 14.3, Problem 49E

Chapter 14.3, Problem 51E

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The velocity vector and position vector, for t ≥ 0 .

Solution Summary: The author explains that the velocity vector and position vector of an object moving in three dimensional space have the acceleration vector a(t)=langle t.