   Chapter 13.4, Problem 38E

Chapter
Section
Textbook Problem

# Find the tangential and normal components of the acceleration vector.38. r ( t ) = 2 t 2 i + ( 2 3 t 3 − 2 t ) j

To determine

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

Explanation

Given data:

r(t)=2t2i+(23t32t)j

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 2t2i+(23t32t)j for r(t) ,

r(t)=ddt[2t2i+(23t32t)j]=ddt[(2t2)i]+ddt[(23t32t)j]=2(2t)i+23(3t2)j2j=4ti+2t2j2j

r(t)=4ti+(2t22)j

Find r(t) .

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

Substitute 4ti+(2t22)j for r(t) ,

r(t)=ddt[4ti+(2t22)j]=ddt(4ti)+ddt[(2t22)j]=4i+2(2t)j0j=4i+4tj

Find |r(t)| .

|r(t)|=(4t)2+(2t22)2=(4t)2+(2t2)28t2+4 {(ab)2=a22ab+b2}=16t2+4t48t2+4=4t4+8t2+4

|r(t)|=4(t2+1)2=2(t2+1

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