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Asked by swayamagarwal2114 | 18 Aug, 2022, 09:04: AM
When the satellite revolving around the planet in an orbit of radius R , then we have
![begin mathsize 14px style fraction numerator m space v squared over denominator R end fraction space equals space G space fraction numerator m space M space over denominator R squared end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/aedae5047e6bb54f1fd6024e4e5bfaf6.png)
LHS of above expression is centrpetal force and RHS of above expression
is gravitational force of attraction.
where m is mass of stellite, M is mass of planet, R is radius of orbit , v is speed of stellite
in the orbot and G is universal gravitational constant .
if T is period of revolution then we have
v = ( 2 π R ) / T .............................................(2)
By substituting v from eqn.(2) , we rewrite eqn.(1) after simplification as
![begin mathsize 14px style fraction numerator 4 pi squared R over denominator T squared end fraction space equals space fraction numerator G space M over denominator R squared end fraction space end style](https://images.topperlearning.com/topper/tinymce/cache/8ada32eb9612c5d030d2dacd5c258e0e.png)
RHS of above expression is acceleration of satellite towards planet when satellite is orbiting at radius R
when the planet is completely stopped , it does not have speed in tangetial direction .
Hence due to to gravitational force of attraction it falls towards planet.
Since the force of attraction on satellite depends on distance between satellite and planet ,
acceleration of falling satellite depends on distance r between satellite and planet.
When satellite is at distance r from planet , its acceleration is given as
![begin mathsize 14px style fraction numerator d v over denominator d t end fraction space equals space fraction numerator G space M over denominator r squared end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/c4b1665cc77440b277d3f14bbcd34c6b.png)
where v is the velocity ar distance r . Eqn.(4) can be written as
![begin mathsize 14px style fraction numerator d v over denominator d t end fraction space equals space fraction numerator d v over denominator d r end fraction cross times fraction numerator d r over denominator d t end fraction space equals space v space fraction numerator d v over denominator d r space end fraction space equals space fraction numerator G space M over denominator r squared end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/2778ece340500c2bd87bba93d484dc86.png)
![begin mathsize 14px style v space d v space equals space G space M space fraction numerator d r over denominator r squared end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/13897eb49bd50cb4795a7c2a877ef07d.png)
By integrating both sides of above expression , we get
![begin mathsize 14px style v squared space equals space fraction numerator 2 space G space M over denominator r end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/ff2ed0cef9dc3d1cc3a3713c95efab90.png)
![begin mathsize 14px style v space equals space fraction numerator d r over denominator d t end fraction space equals fraction numerator square root of 2 space G space M end root over denominator square root of r end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/8633f1ceb0ced26f00e546cf0fd36381.png)
![begin mathsize 14px style square root of r space d r space equals space square root of 2 space G space M end root space d t end style](https://images.topperlearning.com/topper/tinymce/cache/3f456c1cac55bdefb6a0c1031d423876.png)
Let us integrate above expression on both sides
![begin mathsize 14px style integral subscript 0 superscript R square root of r space d r space equals space square root of 2 space G space M end root space integral subscript 0 superscript t d t apostrophe end style](https://images.topperlearning.com/topper/tinymce/cache/49aa7a5327efe345b38752f9742b7563.png)
where t is time taken by satellite to reach the planet
we get from above integration
![begin mathsize 14px style 2 over 3 space R to the power of 3 divided by 2 end exponent space equals space square root of 2 space G space M end root space space t end style](https://images.topperlearning.com/topper/tinymce/cache/3ce688eafb1e96ec0ee65800544dbfa6.png)
Using eqn.(3) , above eqn.(5) can be written as
![begin mathsize 14px style 2 over 3 space R to the power of 3 divided by 2 end exponent space equals space square root of 2 space end root space fraction numerator 2 pi R to the power of 3 divided by 2 end exponent over denominator T end fraction space space t end style](https://images.topperlearning.com/topper/tinymce/cache/2692b0b131ceabde762538c300cd2405.png)
Hence time t is simplified as
![begin mathsize 14px style t space equals space fraction numerator T over denominator 3 square root of 2 space pi end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/17fb2bc6a3e4cb19fccb710ed7ac77bd.png)
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