At what distance from the earth will a spacecraft traveling directly from the earth to the moon experience zero net force because the earth and moon pull with equal and opposite forces?

346, 348 Km.

Explanation:

In the free space, the only forces acting on the spacecraft, are the attraction forces due to gravity exerted at once by Earth and Moon.

At the exact point where both forces are equal and opposite each other (this means that the spacecraft must be somewhere between them), according to Newton's Law of Universal Gravitation, it must be satisfied the following relationship:

Fsp = Fmsp     ⇒   G.me.ms/res² = G.mm.ms/(rme-res)², where:

G= Universal Gravitation Constant

me= mass of the Earth

mm= mass of the Moon ≅ 0.012 me

res= distance from the earth to the spacecraft

rme= distance between Earth and Moon = 384, 400 Km.

Replacing for the different constants, and simplifying equal terms, we can write the following proportion:

1/res² = 0.012/(rme-res)²

Solving the resultant 2nd degree equation for res, we get (after simplifying):

res = 1 ± √0.012 / 0.988 ⇒res = 0.901 res= 384,400 Km . 0.901 =

res = 346,448 Km.

6 cos 34, 6 sin 34 trig