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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

..................................... (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

.................................(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

.....................(4)
where v is the velocity ar distance r . Eqn.(4) can be written as

By integrating both sides of above expression , we get

Let us integrate above expression on both sides

where t is time taken by satellite to reach the planet

we get from above integration

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

Using eqn.(3) , above eqn.(5) can be written as

Hence time t is simplified as

Answered by Thiyagarajan K | 18 Aug, 2022, 02:46: PM
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Asked by swayamagarwal2114 | 18 Aug, 2022, 09:04: AM