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this derivation is based on energy conservation ....
when satellite is at earth surface then
total energy = PE + KE
PE = -GMm/R , (KE)i = ? (M is mass of earth )
TE = (KE)i - GMm/R .....1 (R is radius of earth)
when satellite escapes from earths gravitational field then it reaches to infinity ...
total energy finally = TE = KE + PE
PE = GMm/r ( r is infinity )
PE = 0 , (KE)f = ?
TE = (KE)f + 0 = (KE)f .............2
since energy is conserved so eq 1 equals eq 2
-GMm/R + (KE)i = (KE)f
(KE)i = GMm/R + (KE)f .................3
from this eq we can see that if (KE)f is 0 then (KE) intial is minimum
value of energy should be given to satellite so that it escapes out...
now , (KE)i = mv2/2 , (KE)f = 0 , putting these values in eq 3 we get
v = (2GM/R)1/2
this is the required result...
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