Question(s): The Earth’s moon is moving away from Earth by a few centimeters a year. Will it break free from Earth’s gravitational influence before our Sun turns into a red giant and fries the inner planets of our solar system?
Purely a hypothetical question; if the Moon was to break away from the gravitational pull of Earth now, would it be drawn into its own orbit around the Sun or get drawn into the Sun or get flung out of the solar system all together. – David
Answer(s): Laser ranging measurements of the change in the distance from the Earth to the Moon tell us that the Moon is moving away from the Earth at a rate of about 3.78 cm per year. Calculations of the evolution of the Earth/Moon system tell us that with this rate of separation that in about 15 billion years the Moon will stop moving away from the Earth. Now, our Sun is expected to enter its Red Giant phase in about 6 to 7 billion years. So, the answer to your first question is that the Sun will engulf the inner planets as its outer layers expand during its Red Giant phase before the Moon will stop moving away from the Earth.
As for your second question, the fate of the Moon after being extracted from its orbit around the Earth depends upon exactly what the mechanism was that caused it to be removed from the Earth/Moon system. In general, though, I suspect that if the Moon broke away from the Earth it would end-up drawn into the Sun.