a space capsule is travelling between the earth and the moon.?

find the distance from the earth at which it is subject to zero gravitational force.(consider only the gravitational fields of the earth and the moon.) mass of the earth=6.0*10^24kg
mass of the moon=7.4*10^22kg
Distance between the centres of the earth and the moon=3.8*10^8 m

Well, if you are x distance away from the Earth, you have two forces acting on the space capsule, in opposite directions. One, from the Earth, is

F(earth) = -GEm/x² where E is the mass of the Earth, and

F(moon) = GMm/(d-x)², where M is the mass of the moon and d is the distance between the centers of mass.

To find the zero gravity point, you need to find where the sum of these two forces is zero. So, you solve for it:

GMm/(d-x)² - GEm/x² = 0
GMm/(d-x)² = GEm/x²
M/(d-x)² = E/x²
Mx² = E(d-x)²
√(M)x = √(E) (d-x);
(√(M) + √(E))x = √(E) d
x = √(E)d/(√(M) + √(E))

x = √(6.0*10^24kg)*3.8*10^8m/(√(6.0*10^24kg) + √(7.4*10^22kg)) = 3.4 * 10^8m.

So, the space capsule needs to be 3.4*10^8 meters away from the Earth, which is much closer to the moon than the Earth.