Chapter 12.5, Problem 89E

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

Chapter
Section

### Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem
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# Kepler舗s Laws In Exercises 87-94, you are asked to verify Kepler舗s Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector- valued function r. Let r = r ‖ , let G represent the universal gravitational constant, let M represent the mass of the sun. and let m represent the mass of the planet.Prove that r ⋅ r ' = r d r d t

To determine

To Prove: r.r'=rdrdt.

Explanation

Given:

The vector valued function along which each planet moves in an orbit is

r=r

Formula Used:

r=[x(t)]2+[y(t)]2+[z(t)]2

Proof:

Let r=x(t)i^+y(t)j^+z(t)k^Then r=r=[x(t)]2+[y(t)]2+[z(t)]2and r''=x'(t)i^+y'(t)j^+z'(t)k  So, r(drdt)=[x(t)]2+[y(t)]2+[z(t)2]

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