Question

A satellite which is revolving around the earth has a minimum distance from earth equal to \[{r_1}\] and maximum distance of \[{r_2}\], then the time period of the satellite will be?

Answer 1

Hint: Follow Kelper’s law of Periods where \[{T^2}\] is proportional to \[{a^3}\] .
Here, “a” refers to the semi-major axis.

Formula Used:
1. \[{T^2}\] is proportional to \[{a^3}\]
\[ \Rightarrow \]\[{T^2} = \dfrac{{2\pi {a^3}}}{{GM}}\]
Where,
T= Time period of the satellite
a= semi-major axis
G= gravitational constant
M= mass of the earth (in this case)
Here, \[\dfrac{{2\pi }}{{GM}}\] is the constant of proportionality
2. \[a = \dfrac{{{r_1} + {r_2}}}{2}\]
Where,
\[{r_1}\] = Minimum distance covered
\[{r_2}\] = Maximum distance covered

Complete step by step answer:
Kepler’s gave three laws that describe the movement of planets around the sun .
Among this, the third law of Kepler, also known as the Law of periods states that:
The square of time period of revolution of the planet is proportional to the cube of semi major axis of the ellipse traced by the planet.
The above law can be formulated as follows:
\[{T^2}\] is proportional to \[{a^3}\]
This equation is modified to :
\[{T^2} = \dfrac{{2\pi {a^3}}}{{GM}}\] - (i)
Where,
T= Time period of the satellite
A= semi-major axis
G= gravitational constant
M= mass of the earth (in this case)
Here, \[\dfrac{{2\pi }}{{GM}}\] is the constant of proportionality.
So: \[T = \dfrac{{2\pi {a^{\dfrac{3}{2}}}}}{{GM}}\]

Now, we know the semi-major axis of an elliptical path is the arithmetic mean of the minimum and the maximum distance covered by the planet.
Therefore, in this case:
\[{r_1}\] = Minimum distance covered
\[{r_2}\] = Maximum distance covered
So,
\[a = \dfrac{{{r_1} + {r_2}}}{2}\]
Putting the value of a equation (i) modifies to:
\[T = \dfrac{{2\pi {{\dfrac{{({r_1} + {r_2})}}{2}}^{\dfrac{3}{2}}}}}{{GM}}\]
Thus, this is the value of the time period of the satellite.

Note: The semi-major axis of an elliptical path is the arithmetic mean of the minimum and the maximum distance covered by the planet.
 The semi-minor axis is the geometric mean of minimum and the maximum distance covered by the planet, these two must not be confused.