Explanation Given: The circular orbit r ( t ) = 〈 R cos ( ω t ) , R sin ( ω t ) 〉 . The relationship, ω 2 = k R − 3 Formula used: The differential equation referred in the question. r ' ' ( t ) = − k r ( t ) ‖ r ( t ) ‖ 3 Calculation: We have been given the equation for circular orbit as: r ( t ) = 〈 R cos ( ω t ) , R sin ( ω t ) 〉 Let us figure out its first and second derivatives. ⇒ r ' ( t ) = 〈 − R ω sin ( ω t ) , R ω cos ( ω t ) 〉 ⇒ r ' ' ( t ) = 〈 − R ω 2 cos ( ω t ) , − R ω 2 sin ( ω t ) 〉 Let us now plug all the expressions on both the sides of the equation below: ⇒ 〈 − R ω 2 cos ( ω t ) , − R ω 2 sin ( ω t ) 〉 = − k 〈 R cos ( ω ...

4th Edition

ISBN: 9781319323394

Author: Rogawski

Publisher: MAC HIGHER

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 14.6, Problem 8E

To determine

To verify:

That the given circular orbit satisfies the differential equation $r'' (t) = - \frac{k}{{‖ r (t) ‖}^{2}} e_{r} = - \frac{k r (t)}{{‖ r (t) ‖}^{3}}$ . Then use it to deduce Kepler’s Second Law.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.6, Problem 7E

Chapter 14.6, Problem 9E

Students have asked these similar questions

4. Use method of separation of variable to solve the following wave equation მłu J²u subject to u(0,t) =0, for t> 0, u(л,t) = 0, for t> 0, = t> 0, at² ax²' u(x, 0) = 0, 0.01 x, ut(x, 0) = Π 0.01 (π-x), 0

Solve the following heat equation by method of separation variables: ди = at subject to u(0,t) =0, for -16024 ძx2 • t>0, 0 0, ux (4,t) = 0, for t> 0, u(x, 0) = (x-3, \-1, 0 < x ≤2 2≤ x ≤ 4.

ex 5. important aspects. Graph f(x)=lnx. Be sure to make your graph big enough to easily read (use the space given.) Label all 6 33

Chapter 14 Solutions

CALCULUS W/SAPLING ACCESS >IC<

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 6PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

To verify: That the given circular orbit satisfies the differential equation r ' ' ( t ) = − k ‖ r ( t ) ‖ 2 e r = − k r ( t ) ‖ r ( t ) ‖ 3 . Then use it to deduce Kepler’s Second Law.

Concept explainers

Videos

To verify: That the given circular orbit satisfies the differential equation r''(t)=−k‖r(t)‖2er=−kr(t)‖r(t)‖3. Then use it to deduce Kepler’s Second Law.

Want to see the full answer?

Chapter 14 Solutions

To verify:

That the given circular orbit satisfies the differential equation $r'' (t) = - \frac{k}{{‖ r (t) ‖}^{2}} e_{r} = - \frac{k r (t)}{{‖ r (t) ‖}^{3}}$ . Then use it to deduce Kepler’s Second Law.