   Chapter 13.4, Problem 37E

Chapter
Section
Textbook Problem

# Find the tangential and normal components of the acceleration vector.37. r(t) = (t2 + 1) i + t3 j, t ≥ 0

To determine

To find: The tangential components of the acceleration vector and normal components of the acceleration vector.

Explanation

Given data:

r(t)=(t2+1)i+t3j

Formula used:

Write the expression for tangential component.

aT=r(t)r(t)|r(t)| (1)

Write the expression for normal component.

aN=|r(t)×r(t)||r(t)| (2)

Find r(t) .

r(t)=ddt[r(t)]

Substitute (t2+1)i+t3j for r(t) ,

r(t)=ddt[(t2+1)i+t3j]=ddt[(t2+1)i]+ddt[t3j]=2ti+3t2j

Find r(t) .

r(t)=ddt[r(t)]

Substitute 2ti+3t2j for r(t) ,

r(t)=ddt(2ti+3t2j)=ddt(2ti)+ddt(3t2j)=2i+3(2t)j=2i+6tj

Find |r(t)| .

|r(t)|=(2t)2+(3t2)2=4t2+9t4=t4+9t2

Substitute 2ti+3t2j for r(t) , 2i+6tj for r(t) and t4+9t2

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