What is the distance from the moon to the point between Earth and the Moon where the gravitational pulls of Earth and Moon are equal? The mass of Earth is 5.97 × 10 24 kg, the mass of the Moon is 7.35 × 10 22 kg, the distance between Earth and the Moon is 3.84 × 10 8 m, and

RE = 3.456*10⁸ m

Explanation:

Given

G = 6.67*10⁻¹¹ N*m²/Kg²

ME = 5.97*10²⁴ Kg

MM = 7.35*10²² Kg

R = 3.84*10⁸ m

The forces acting on a mass m from the Earth (mass ME) and the moon (mass MM) are:

FE = G*m*ME/RE²    and   FM = G*m*MM/RM²

where RE and RM are distances from the mass m to the Earth and the Moon.

From the given conditions:

FE = FM   and  

RE + RM = R =3.84*10⁸ m (the distance between the Earth and the Moon).

Then:  

ME/RE² = MM/RM²    ⇒  (R-RE)²/RE² = MM/ME    

⇒   (R/RE) - 1 = √(MM/ME)      

⇒ RE = R / (1 + √(MM/ME)) =

⇒ RE = 3.84*10⁸ m / (1 + √(7.35*10²² Kg / 5.97*10²⁴ Kg)) = 3.456*10⁸ m.

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^^something like that^^^