   Chapter 13.2, Problem 55E

Chapter
Section
Textbook Problem

# If r(t) ≠ 0, show that d d t | r ( t ) | = 1 | r ( t ) | r ( t ) ⋅ r ' ( t ) . [Hint: |r(t)| 2 = r(t)·r(t)]

To determine

To show: The expression ddt|r(t)| is equals to the expression 1|r(t)|r(t)r(t) when the vector r(t)0 .

Explanation

Given data:

r(t)0

Write the expression to compute |r(t)|2 as follows.

|r(t)|2=r(t)r(t) (1)

Rewrite the expression as follows.

|r(t)|=[r(t)r(t)]12

Differentiate on both sides of the expression with respect to t .

ddt|r(t)|=ddt[r(t)r(t)]12

Use the following formulae to compute the expression.

ddt[r(t)r(t)]n=n[r(t)r(t)]n1ddt[r(t)r(t)]ddt[r(t)r(t)]=|r(t)r(t)+r(t)r(t)|ab=ba

Compute the expression ddt|r(t)|=ddt[r(t)r(t)]12 by using the formulae as follows.

ddt|r(t)|=12[r(t)r(t)]121ddt[r(t)r(t)]=12[r(t)r(t)]12[r′</

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