Final answer:
Kepler's first law, stating that planets orbit in ellipses with the Sun at one focus, is based on gravity being an inverse-square law. If gravity followed an inverse cube law, Kepler's first law would not hold, as the orbits would be unstable and non-elliptical. Kepler's third law would remain valid for objects like the Moon, given the same mass and distance from the collapsed Earth.
Step-by-step explanation:
The question would Kepler's first law be true if gravity was an inverse cube law falls within the realm of physics, considering the laws of planetary motion and gravitational forces. Kepler's first law, also known as the law of ellipses, states that the orbit of a planet is an ellipse with the Sun at one of the two foci. This law is predicated on gravity being an inverse-square law, as articulated by Newton's law of universal gravitation (F = Gm₁m₂/r²).
If gravity followed an inverse cube law (F ≈ m₁m₂/r³) instead, the stable orbits predicted by Kepler's laws would not exist, as the gravitational force would decrease more rapidly with distance than in an inverse-square law. Therefore, orbits would not be elliptical but would follow different patterns altogether and likely lead to unstable planetary motions.
In relation to Kepler's third law, it states that the square of a planet's orbital period is directly proportional to the cube of the semimajor axis of its orbit. This law assumes an inverse-square law for gravity. If Earth collapsed into a dense object like a black hole, but with the same mass, the Moon's period of revolution would remain unchanged since the gravitational force at the Moon's orbit would be unaffected by Earth's new size.