Four billion years into the future, the Moon will be about twice the distance to the Earth.  How much higher/lower will the lunar tides be?_______x the present lunar tide heights   (numerical answer)

The height of the tides on Earth is determined by the gravitational pull of the moon. The gravitational force between two objects is given by the formula:

F = G * (m1 * m2) / r^2

where:

• F is the force between the objects,
• G is the gravitational constant,
• m1 and m2 are the masses of the objects, and
• r is the distance between the centers of the two objects.

In this case, the masses of the Earth and the Moon won't change, so we can ignore those. The gravitational constant also won't change. So the change in the force (and thus the change in the tide height) is determined by the change in the distance.

If the distance doubles (r becomes 2r), then the force becomes:

F_new = G * (m1 * m2) / (2r)^2 = G * (m1 * m2) / 4r^2 = F_old / 4

So the gravitational force (and thus the tide height) will be 1/4 of what it is now. Therefore, the lunar tides will be 0.25 times the present lunar tide heights.