Suppose that the Earth were moved to a distance of 0.25 AU from the Sun. How much
stronger or weaker would the Sun's gravitational pull on the Earth? Explain.

It increases by 16 times

Explanation:

The magnitude of the gravitational force between two objects is given by the equation

where

G is the gravitational constant

m1, m2 are the masses of the two objects

r is their separation

In this problem, we call:

is the Earth's mass

is the Sun's mass

is the initial distance Earth-Sun

So the gravitational force between the two objects is

Later, the Earth is moved to a distance of

r' = 0.25 AU

which is equivalent to write

from the Sun.

Therefore, the new gravitational force will be:

So, the gravitational force increases by a factor of 16.