Question

At what distance from the earth is the gravitational pull of the sun equal to the gravitational pull of the earth?

Answer 1

The condition for the gravitationalpull of the sun to be equal to the Earth's one is
F1=F2
GmM1/(r1^2)= GmM2/(r2^2)

where r1 is distance to Sun, M1 is mass of the Sun, r2 is distance to Earth and
M2 mass of the Earth.
From this we can find, what relation between distances toEarth and to the Sun should be

r1/r2 = sqrt(M1/M2)
knowing also, that their difference should give distancebetween Sun and Earth
r1^2-r2^2 = R^2
we can find r1:
r2= r1/sqrt(M1/M2)

r1^2+r1^2/(M1/M2) = R^2
r1^2 = R^2/(1+M1/M2)
r1 = R/sqrt(1+M1/M2)
r1 = 149*10^9 m from Earth