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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 x 10^24 kg, the mass of the Moon is 7.35 x 10^22 kg, the distance between Earth and the Moon is 3.84 x 10^8 m, and >G= 6.67x10^-11N x m^2/kg^2
A)3.83 x 10^6 mB)3.83 x 10^7 mC)4.69 x 10^6 mD)4.69 x 10^7 mE)3.45 x 10^8 m

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Answer:

the point between Earth and the Moon where the gravitational pulls of Earth and Moon are equal is E)3.45 × 10⁸ m

Explanation:

The force that the Earth exerts on a mass m is

F_e = (G M_e m) / R_e²

where

  • G is the universal gravitational constant
  • M_e is the mass of Earth
  • R_e is the radius of Earth

The force that the Moon exerts on a mass m is

F_m = (G M_m m) / R_m²

where

  • G is the universal gravitational constant
  • M_m is the mass of the Moon
  • R_m is the radius of the Moon

Therefore, the point where the gravitational pulls of Earth and Moon are equal is:

F_e = F_m

R_e + R_m = R = 3.84×10⁸ m

Thus,

(G M_e m) / R_e² = (G M_m m) / R_m²

M_e / R_e² = M_m / (R - R_e²)

(R - R_e²) / R_e² = M_m / M_e

(R - R_e) / R_e = (M_m / M_e)^1/2

R_e(R/R_e -1) / R_e = (M_m / M_e)^1/2

R/ R_e = (M_m / M_e)^1/2 + 1

R_e = R / [(M_m / M_e)^1/2 + 1]

R_e = (3.84×10⁸ m) / [(7.35 x 10²² kg / 5.97 x 10²⁴ kg )^1/2 + 1]

R_e = 3.45 × 10⁸ m

Therefore, the  point between Earth and the Moon where the gravitational pulls of Earth and Moon are equal is 3.45 × 10⁸ m.

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