A recent paper by Alexander Klotz of McGill University, Montreal, has stirred a bit of interest in the pop-science press. The paper looks at the old question of the time taken to fall through a hole passing through the centre of the Earth, and back up to the surface at the opposite point on the Earth’s surface. This calculation makes the usual assumptions of a perfect vacuum, and zero friction, but instead of the usual simplification of assuming the Earth to be of constant density, it allows for the actual estimated varying density along the length of the hole, and finds that the transit time would be reduced from 44 minutes to just over 38 minutes.
There is a good article on the paper at How long would it take you to fall through Earth?, and a rather more excitable one at How Long Would It Take To Fall Through The Center Of The Earth?
The latter article suggests that “Klotz went where no physicist has gone before”. I suppose that is possible, but I know that at least one engineer has looked at the question before, I have on this very blog:
I came up with a time of 38 minutes and 57 seconds, compared with 38 minutes and 11 seconds found by Klotz.
Another surprising finding was that if a constant acceleration of 9.81 m/s2 is assumed all the way to the centre, then -9.81 m/s2 all the way back up to the surface, the transit time is only reduced to 38 minutes. The graph below shows what is happening; the effect of increasing density is that the acceleration does remain close to constant over much of the journey, and it is only in the final stages that the velocity assuming constant acceleration becomes significantly faster than the velocity based on the actual (estimated) density.
More links on the topic of holes through the middle of the Earth: