Question

At what point (in km from Earth) between Earth and Saturn is the net force of gravity on a body by both Earth and Saturn exactly zero?

Answer 1

see below

Explanation:

Gravity force is never zero, but at a certain distance the two force can be the same:
When the two force are the same the force between earth and body, FG1:
`F_(G1)= K_G xx (M_E xx M_B)/ d_(EB)^2`
where KG is the universal gravitational constant, ME and MG the mass of Earth and of the body and d the distance,
is equal at the force between Saturn and body FG2:
`F_(G2)= K_G xx (M_S xx M_B)/ d_(SB)^2`

so you have:
`K_G xx (M_E xx M_B)/ d_(EB)^2=K_G xx (M_S xx M_B)/ d_(SB)^2`
simplifying the same terms, you have
(1) `(M_E )/ d_(EB)^2= (M_S )/ d_(SB)^2`
Now you can early find on internet the mass of Earth and of Saturn, but not the distance d that changes continuosly.
In a certain instant you have
`d_EB+ d_SB = D`
From this equation you can obtain `d_EB = D- d_SB` and put it in equation (1) solving your problem