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Asked by swayamagarwal2114 | 18 Aug, 2022, 09:04: AM
answered-by-expert Expert Answer
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 ..................................... (1)
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 .................................(3)
 
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.....................(4)
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
 
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
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
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

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
 
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
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  ..............................(5 )
 
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
 
 
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

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