We first need to establish the scale factor we would have to shrink the Earth by in order to reduce it to the size of a squash ball.
The Earth is 7,926 miles in diameter at the equator, and a regulation squash ball has a diameter of 1.7 inches. This means that to shrink the Earth down to the size of a squash ball, its size would have to be multiplied by a scale factor of 3.45 x 10-9.
To compare the smoothness of the two surfaces, we need to know the variation of the surface, that is, the difference between the highest and lowest points.
For a squash ball, this is a simple process, because there are very few areas where the surface is higher than the average, but there are many small indentations or depressions.
Since these depressions are roughly 0.004 inches in depth, the variation in surface height can be taken to be roughly 0.004 inches.
For the Earth, the lowest point below the surface is in the Mariana Trench, which is 36,200 feet below sea level at its deepest point, known as the Challenger Deep. The highest point is, of course, the summit of Mount Everest, which is estimated at 29,029 feet above sea level.
Therefore, the variation in the height of the Earth’s surface is 65,229 feet.
If we scaled the Earth down to the size of a squash ball, using the scale factor calculated above, the variation of its surface would be 2.25 x 104 feet, or 0.0027 inches.
This figure is in fact about two-thirds of the figure for the squash ball, so what your correspondent heard is actually true: if Earth were scaled down to this size it would indeed be smoother than the average regulation squash ball.
Now for the second part of the question. The lack of any raised areas on the squash ball’s surface means that there are in fact mostly depressions or indentations in the surface.
So if a squash ball were scaled up to the size of the Earth, there would be no mountains as such.
There would, however, be a lot of large craters. In fact, if we scaled up the indentations in the ball’s surface we would end up with some immense depressions that were almost 18 miles deep.
If these depressions were ocean trenches like those that are found on the Earth’s surface, they would penetrate the 3.7-mile-thick oceanic crust, and extend right through the Mohorovicic discontinuity where the crust meets the mantle and well into the mantle beneath.
These craters would not only be deep but could be anything up to about 37 miles wide.
If the mass of the Earth were crushed down to the size of a squash or racketball, then it would be dense enough to be either neutronium or a black hole.
In the case of neutronium, the gravity at the surface would be in the order of at least a million times the gravity you are feeling now, more than enough to smooth out any irregularities in the surface. In the case of a black hole there wouldn’t be a surface, just an event horizon which would be smooth.
Expanding a ball to the size of the Earth is somewhat different. If we assume that the ball is made mainly of carbon atoms and weighs 2 pounds, then there would only be around 352 atoms in each cubic inch.
This is actually less dense than the Earth’s upper atmosphere on the edge of outer space.