A rocket is fired from the earth towards the moon. At what distance from the moon is the gravitational force on the rocket is zero. Mass of earth =610^24 kg, Mass of moon =7.410^22 kg and orbital radius =3.81,^8 m. Neglect the effect of the sun and other planets. ( answer : 3.810^7 m)
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A point between Earth and Moon, at which the net gravitational force is 0. This is the Lagrangian point between Earth and Moon.
Let the point P be at a distance x * D from Moon. D is the distance between Earth and Moon. x is a fraction, 0 < x < 1.
Mass of Earth = Me = 6.10 * 10²⁴ Kg
Mass of Moon = Mm = 7.410 * 10²² kg
D = 3.81 * 10⁸ m
Mass of a third body = m.
Newton's Gravitational force due to Earth at point P = force due to Moon at P
G Me m / (D - x D)² = G Mm * m/ (xD)²
x Me D² = (1-x)² D² Mm
x² * 6.10 * 10²⁴ = (1- x)² * 7.410 * 10²²
82.321 x² = 1 + x² - 2 x
x² * 81.321 + 2 x - 1 = 0
x = 0.0993
Distance of the satellite from Moon = x D
= 3.78 * 10⁷ m
Let the point P be at a distance x * D from Moon. D is the distance between Earth and Moon. x is a fraction, 0 < x < 1.
Mass of Earth = Me = 6.10 * 10²⁴ Kg
Mass of Moon = Mm = 7.410 * 10²² kg
D = 3.81 * 10⁸ m
Mass of a third body = m.
Newton's Gravitational force due to Earth at point P = force due to Moon at P
G Me m / (D - x D)² = G Mm * m/ (xD)²
x Me D² = (1-x)² D² Mm
x² * 6.10 * 10²⁴ = (1- x)² * 7.410 * 10²²
82.321 x² = 1 + x² - 2 x
x² * 81.321 + 2 x - 1 = 0
x = 0.0993
Distance of the satellite from Moon = x D
= 3.78 * 10⁷ m
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The answer is that 3.78*10^7
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