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A Satellite is in an elliptic orbit around the earth with an aphelion of $6 \mathrm{R}$ and
perihelion of $2 \mathrm{R}$. Find the eccentricity of the orbit.
A. $\dfrac{1}{6}$
B. $\dfrac{1}{3}$
C. $\dfrac{1}{2}$
D. $\dfrac{1}{4}$

Last updated date: 01st Mar 2024
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IVSAT 2024
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Hint: We know that the maximum distance of load from the centre of column, such that if load acts within this distance there is no tension in the column. The maximum distance is called the Limit of eccentricity. Eccentricity measures how much the shape of Earth's orbit departs from a perfect circle. These variations affect the distance between Earth and the Sun. Because variations in Earth's eccentricity are fairly small, they're a relatively minor factor in annual seasonal climate variations. The orbital eccentricity (or eccentricity) is a measure of how much an elliptical orbit is 'squashed'. Elliptical orbits with increasing eccentricity from $\mathrm{e}=0$ (a circle) to $\mathrm{e}=0.95 .$ For a fixed value of the semi-major axis, as the eccentricity increases, both the semi-minor axis and perihelion distance decrease.

Complete step by step answer
We know that eccentricity is a measure of how an orbit deviates from circular. A perfectly circular orbit has an eccentricity of zero; higher numbers indicate more elliptical orbits. Neptune, Venus, and Earth are the planets in our solar system with the least eccentric orbits. In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape. The eccentricity of an ellipse which is not a circle is greater than zero but less than 1. The eccentricity of a parabola is 1. The eccentricity of a hyperbola is greater than 1.
Let us consider that $\mathrm{r}_{\mathrm{a}}$ and $\mathrm{r}_{\mathrm{p}}$ denote distance of aphelion and perihelion of the
elliptical orbit of the satellite.
$\mathrm{r}_{\mathrm{a}}=\mathrm{a}(1+\mathrm{e})$ and $\mathrm{r}_{\mathrm{p}}=(1-\mathrm{e})(\mathrm{a}$ is semi major axis of ellipse)
As $\mathrm{r}_{\mathrm{a}}=\mathrm{bl}$ and $\mathrm{r}_{\mathrm{p}}=2 \mathrm{R}$
$\dfrac{\mathrm{a}(1+\mathrm{e})}{\mathrm{a}(1-\mathrm{e})}=\dfrac{6 \mathrm{R}}{2 \mathrm{R}}=3$ where $\mathrm{e}=\dfrac{1}{2}$
So, eccentricity $=\dfrac{1}{2}$

So, option C is correct.

Note: We know that gravitation is the attractive force existing between any two objects that have mass. The force of gravitation pulls objects together. Gravity is the gravitational force that occurs between the earth and other bodies. Gravity is the force acting to pull objects toward the earth. Gravity, also called gravitation, in mechanics, the universal force of attraction acting between all matter. On Earth all bodies have a weight, or downward force of gravity, proportional to their mass, which Earth's mass exerts on them. Gravity is measured by the acceleration that it gives to freely falling objects.