Explanation Given: Earth revolves around the sun in an approximately circular orbit with radius r = a , completing a revolution in the time T = 2 π ( a 3 / G M ) 1 / 2 , which is one Earth year; here M is the mass of the sun and G is the universal gravitational constant. The gravitational force of the sun on Earth is given by G M m / r 2 , where m is the mass of Earth. Therefore, if Earth “stood still,” losing its accordance with Newton’s second law: m d 2 r d t 2 = − G M m r 2 . Approach: The Energy Integral Lemma: Let y ( t ) be a solution to the differential equation y ″ = f ( y ) where f ( y ) is a continuous function that does not depend on y ′ or the independent variable t . Let F ( y ) be an infinite integral of f ( y ) , that is, f ( y ) = d d y F ( y ) Then the quantity, E ( t ) = 1 2 ( y ′ ( t ) ) 2 − F ( y ( t ) ) is constant; that is, d d t E ( t ) = 0 . Calculation: Rearrange the relation, m d 2 r d t 2 = − G M m r 2 . d 2 r d t 2 = − G M r 2 Apply the energy integral lemma, F ( r ) = ∫ − G M r 2 = G M r ( 1 ) Thus, k = 1 2 ( r ′ ( t ) ) 2 − G M r ( t ) … ( 1 ) Apply the initial conditions r ′ ( 0 ) = 0 , r ( 0 ) = a in equation ( 1 ) . k = 0 − G M a = − G M a Substitute − G M a for k in equation ( 1 ) . − G M a = 1 2 ( r ′ ( t ) ) 2 − G M r ( t ) 1 2 ( r ′ ( t ) ) 2 = G M r ( t ) − G M a ( d r d t ) 2 = 2 G M ( 1 r − 1 a ) d r d t = ± 2 G M ( 1 r − 1 a ) As r is decreasing so negative sign will be considered, d r d t = − 2 G M ( 1 r − 1 a ) This implies, − d r 1 r − 1 a = 2 G M d t Integrate on both sides, − ∫ d r ( 1 r − 1 a ) = 2 G M ∫ d t ... ( 2 ) Use the method of substitution, substitute u 2 = 1 r − 1 a . Differentiate u 2 = 1 r − 1 a with respect to u . 2 u d u = − r 2 d r As r 2 = a 2 ( a u 2 + 1 ) 2 , d r = − 2 u a 2 ( a u 2 + 1 ) 2 d u Substitute d r = − 2 u a 2 ( a u 2 + 1 ) 2 d u in equation ( 2 ) . − 2 a 2 ∫ d u ( u 2 a + 1 ) 2 = 2 G M ( t ) + C .... ( 3 ) Again use the method of substitution, substitute u = 1 a tan θ

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

See similar textbooks

Videos

Question

Chapter 4.8, Problem 17E

To determine

The fraction of the normal year T that would require by Earth to splash into the sun.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.8, Problem 16E

Chapter 4.9, Problem 1E

Students have asked these similar questions

For the system consisting of the three planes:plane 1: -4x + 4y - 2z = -8plane 2: 2x + 2y + 4z = 20plane 3: -2x - 3y + z = -1a) Are any of the planes parallel and/or coincident? Justify your answer.b) Determine if the normals are coplanar. What does this tell you about the system?c) Solve the system if possible. Show a complete solution (do not use matrix operations). Classify the system using the terms: consistent, inconsistent, dependent and/or independent.

Open your tool box and find geometric methods, symmetries of even and odd functions and the evaluation theorem. Use these to calculate the following definite integrals. Note that you should not use Riemann sums for this problem. (a) (4 pts) (b) (2 pts) 3 S³ 0 3-x+9-dz x3 + sin(x) x4 + cos(x) dx (c) (4 pts) L 1-|x|dx

A movie studio wishes to determine the relationship between the revenue generated from the streaming of comedies and the revenue generated from the theatrical release of such movies. The studio has the following bivariate data from a sample of fifteen comedies released over the past five years. These data give the revenue x from theatrical release (in millions of dollars) and the revenue y from streaming (in millions of dollars) for each of the fifteen movies. The data are displayed in the Figure 1 scatter plot. Theater revenue, x Streaming revenue, y (in millions of (in millions of dollars) dollars) 13.2 10.3 62.6 10.4 20.8 5.1 36.7 13.3 44.6 7.2 65.9 10.3 49.4 15.7 31.5 4.5 14.6 2.5 26.0 8.8 28.1 11.5 26.1 7.7 28.2 2.8 60.7 16.4 6.7 1.9 Streaming revenue (in millions of dollars) 18+ 16+ 14 12+ xx 10+ 8+ 6+ 2- 0 10 20 30 40 50 60 70 Theater revenue (in millions of dollars) Figure 1 Send data to calculator Send data to Excel The least-squares regression line for these data has a slope…

Chapter 4 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 4.1 - Verify that for b=0 and Fext(t)=0, equation (3)...Ch. 4.1 - If Fext(t)=0, equation (3) becomes my+by+ky=0. For...Ch. 4.1 - Show that if Fext(t)=0, m=1, k=9, and b=6, then...Ch. 4.1 - Prob. 4E Ch. 4.1 - Verify that the exponentially damped sinusoid...Ch. 4.1 - An external force F(t)=2cos2t is applied to a...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - In Problems 79, find a synchronous solution of the...Ch. 4.1 - Undamped oscillators that are driven at resonance...

Knowledge Booster

Learn more about

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

The fraction of the normal year T that would require by Earth to splash into the sun.

Solution Summary: The author calculates the tion of the normal year T that would require Earth to splash into the sun.

Videos

The fraction of the normal year T that would require by Earth to splash into the sun.

Want to see the full answer?

Chapter 4 Solutions