On the way from a planet to a moon, astronauts reach a point where that moon’s gravitational pull transitions from being weaker than that of the planet to being stronger than that of the planet. The masses of the planet and the moon are, respectively, 5.93 × 1024 kg and 7.36×1022 kg. The distance from the center of the planet to the center of the moon is 3.45 × 108 m. Determine the distance of this point from the center of the planet. The value of the universal gravitational constant is 6.67259×10−11 N·m2/kg2.
To determine the distance from the center of the planet to the point where the moon's gravitational pull transitions, we can use the concept of gravitational force and equal gravitational potentials.
First, let's calculate the gravitational potential energy at the surface of the planet and the moon. The gravitational potential energy (PE) is given by the formula:
PE = - G * (m1 * m2) / r
Where G is the universal gravitational constant, m1 and m2 are the masses of the interacting bodies, and r is the distance between their centers.
For the planet:
PE_planet = - (6.67259×10^(-11) N·m²/kg²) * (5.93 × 10²⁴ kg) * (7.36 × 10²² kg) / (3.45 × 10⁸ m)
For the moon:
PE_moon = - (6.67259×10^(-11) N·m²/kg²) * (7.36 × 10²² kg) * (5.93 × 10²⁴ kg) / (3.45 × 10⁸ m)
To find the point where the moon's gravitational pull becomes stronger, we need to find the distance at which these potentials are equal. So, we set PE_planet = PE_moon and solve for r:
- (6.67259×10^(-11) N·m²/kg²) * (5.93 × 10²⁴ kg) * (7.36 × 10²² kg) / (3.45 × 10⁸ m) = - (6.67259×10^(-11) N·m²/kg²) * (7.36 × 10²² kg) * (5.93 × 10²⁴ kg) / r
By canceling out terms and solving for r, we get:
r = (3.45 × 10⁸ m) * (7.36 × 10²² kg) / (5.93 × 10²⁴ kg)
Simplifying further, we find:
r ≈ 4.264 × 10⁶ m
Therefore, the distance from the center of the planet to the point where the moon's gravitational pull transitions is approximately 4.264 × 10⁶ meters.