Aspacecraft is on a journey to the moon. at what point, as measured from the center of the earth, does the gravitational force exerted on the spacecraft by the earth balance that exerted by the moon? this point lies on a line between the centers of the earth and the moon. the distance between the earth and the moon is 3.85 × 108 m, and the mass of the earth is 81.4 times as great as that of the moon.

This distance is measured from the center of the earth   r = 3.4 10⁸ m

Explanation:

The equation for gravitational attraction force is

      F = G m1 m2 / r²

Where g is the universal gravitation constant, m are the masses of the body and r is the distance between them, remember that this force is always attractive

Let's write the sum of force on the ship and place the condition that is balanced

    F1 -F2 = 0

    F1 = F2

Let's write this equation for our case

   G m Me / r² = G m Mm / (r'.)²

The distance r is measured from the center of the earth and the distance r' is measured from the center of the moon,

      r' = 3.85 10⁸ m

Let's simplify and calculate the distance

     Me / r² = Mm / / (3.85 108- r)²

     Me / Mm (3.85 108- r)² = r²

     √ 81.4 (3.85 108 -r) = r

     √ 81.4  3.85 108 = r (1 + √ 81.4)

     34.74 108 = r (10.02)

     r = 34.74 10⁸ / 10.2

     r = 3.4 10⁸ m

This distance is measured from the center of the earth

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