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Asked by swayamagarwal2114 | 18 Aug, 2022, 09:04: AM

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

Answered by Thiyagarajan K | 18 Aug, 2022, 14:46: PM

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