Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 6: Problem 799

The largest and shortest distance of the earth from the sun are \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\) its distance from the sun when it is at the perpendicular to the major axis of the orbit drawn from the sun (A) \(\left[\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right) / 4\right]\) (B) $\left[\left(\mathrm{r}_{1} \mathrm{r}_{2}\right) /\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right)\right]$ (C) \(\left[\left(2 r_{1} r_{2}\right) /\left(r_{1}+r_{2}\right)\right]\) (D) \(\left[\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right) / 3\right]\)

Short Answer

Expert verified
The distance of the Earth from the Sun when it is at the perpendicular to the major axis of the orbit drawn from the Sun is \(\frac{r_1 + r_2}{2}\).

Step by step solution

01

Understand the properties of ellipses

An ellipse has two foci, and the sum of the distances from any point on the ellipse to the two foci is equal to a constant (the major axis length). For the Earth's orbit, the Sun is at one of the foci. Let F1 and F2 be the foci of the ellipse, and let P be any point on the ellipse. Then, the equation for the ellipse is given by: \[PF_1 + PF_2 = constant\] In our problem, F1 is at the Sun, and F2 is at the other focus.
02

Determine the relationship between the major and minor axes and the largest and shortest distances

The largest and shortest distances of the Earth from the Sun, \(r_1\) and \(r_2\), correspond to the endpoints of the major axis of the ellipse. The distance between the foci is \(2c\), where \(c\) is the distance from the center of the ellipse to one of its foci. Using the properties of ellipses, we can relate the major axis length, the minor axis length, and the distance between the foci: \[a^2 = b^2 + c^2\] where \(a\) is half of the major axis length (the semi-major axis) and \(b\) is half of the minor axis length (the semi-minor axis).
03

Find the expression for the distance when Earth is perpendicular to the major axis

When the Earth is at a position perpendicular to the major axis, the distances from the Earth to the foci, \(PF_1\) and \(PF_2\), are equal in length, since the Earth's position lies on the minor axis of the ellipse. Let the distance from the Earth to the Sun at this position be denoted by \(r\). Thus, we can write the equation for the ellipse as: \[2r = r_1 + r_2\] Solving for \(r\), we get: \[r = \frac{r_1 + r_2}{2}\]
04

Compare the obtained expression with the given options

Comparing our obtained expression for \(r\) with the given options, we can see that the correct option is: (A) \(\left[\left(\mathrm{r}_{1}+\mathrm{r}_{2}\right) / 4\right]\) So, the distance of the Earth from the Sun when it is at the perpendicular to the major axis of the orbit drawn from the Sun is \(\frac{r_1 + r_2}{2}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If the value of ' \(\mathrm{g}\) ' acceleration due to gravity, at earth surface is \(10 \mathrm{~ms}^{-2}\). its value in \(\mathrm{ms}^{-2}\) at the center of earth, which is assumed to be a sphere of Radius ' \(\mathrm{R}\) 'meter and uniform density is (A) 5 (B) \(10 / \mathrm{R}\) (C) \(10 / 2 \mathrm{R}\) (D) zero

Two satellites \(\mathrm{A}\) and \(\mathrm{B}\) go round a planet \(\mathrm{p}\) in circular orbits having radii \(4 \mathrm{R}\) and \(\mathrm{R}\) respectively if the speed of the satellite \(\mathrm{A}\) is \(3 \mathrm{~V}\), the speed if satellite \(\mathrm{B}\) will be (A) \(12 \mathrm{~V}\) (B) \(6 \mathrm{~V}\) (C) \(4 / 3 \mathrm{~V}\) (D) \(3 / 2 \mathrm{~V}\)

The period of a satellite in circular orbit around a planet is independent of (A) the mass of the planet (B) the radius of the planet (C) mass of the satellite (D) all the three parameters (A), (B) and (C)

Escape velocity of a body of \(1 \mathrm{~kg}\) on a planet is $100 \mathrm{~ms}^{-1}$. Gravitational potential energy of the body at the planet is \(=\) $\begin{array}{ll}\text { (A) } \overline{-5000} & \text { (B) }-1000\end{array}$ (C) \(-2400\) (D) 5000

Direction (Read the following questions and choose) (A) If both Assertion and Reason are true and the Reason is correct explanation of assertion (B) If both Assertion and Reason are true, but reason is not correct explanation of the Assertion (C) If Assertion is true, but the Reason is false (D) If Assertion is false, but the Reason is true Assertion: The value of acc. due to gravity \((\mathrm{g})\) does not depend upon mass of the body Reason: This follows from \(\mathrm{g}=\left[(\mathrm{GM}) / \mathrm{R}^{2}\right]\), where \(\mathrm{M}\) is mass of planet (earth) and \(\mathrm{R}\) is radius of planet (earth) (a) \(\mathrm{A}\) (b) \(\mathrm{B}\) (c) \(\mathrm{C}\) (d) D
See all solutions

Recommended explanations on English Textbooks

Single Paragraph Essay

Read Explanation

Blog

Read Explanation

English Language Study

Read Explanation

Lexis and Semantics Summary

Read Explanation

Cues and Conventions

Read Explanation

Essay Writing Skills

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free