Yet Even More Marilyn!

Still, there are also times when I simply do not understand her answer; sometimes the wording is confusing and other times is a bit vague. This is the case for her column in the Parade magazine from January 24, 1999. Ms. vS was asked a question that almost every astronomy and physics teacher has been asked at some time: ``If a hole were drilled all the way through the Earth and someone jumped in, what would happen?'' A quick search of USENET posts using www.dejanews.com reveals many such discussions. I have answered a similar question myself on my 1996 Mad Science page.

In a nutshell, the answer is that you would fall down the hole, accelerating (that is moving faster and faster) all the way to the center of the Earth. Then when you passed the center, you would in essence be falling up, and so you would slow down (decelerate) all the way to the top. When you reached the other end of the hole, you would have essentially zero velocity, and then you would fall again, starting your ride all over. This assumes there is no air resistance (which would otherwise put a maximum speed - terminal velocity- on your descent), and that you can ignore the effect of the rotating Earth (which would otherwise have you bumping constantly against the west end of the hole). [Oops! (June 14, 2000) Actually, you'd bump against the east part of the hole, contrary to common sense. Imagine: at the Earth's surface, you are spinning around once at some velocity (you are going all the way around the Earth in a circle every 24 hours). Near the core, that circle is much smaller, so your velocity is much lower. So as you fall, you are moving forward faster then the hole is, so you actually fall ahead it, to the East. This is a lesser known outcome of the Coriolis Effect. It's so little known that I had forgotten about it when I wrote the paragraph above! My thanks to Bad Reader Dr. Neal Rowell for setting me straight.]

To Ms. vS's credit, she gets this answer right, although I think a lot of her phrasing is a bit loose (due to copyright considerations, I cannot quote her article in its entirety here). However, she makes a statement that is, well, odd, as well as incorrect. When explaining the initial fall, she says:

Although gravity would get weaker as you approach the center [of the Earth], your motion would keep you accelerating [the emphasis is mine].

Actually, your motion has nothing to do with it. What is accelerating you is the amount of mass enclosed in the sphere between you and the center of the Earth. In other words, draw a line between the center of the Earth and you. Now make a sphere with that radius centered on the center of the Earth. All the mass in that sphere is what pulls you down (the mass outside that radius-- essentially the mass ``above'' you-- does not contribute to your acceleration). When you start, that sphere encloses the whole Earth. As you get closer to the center, that sphere gets smaller, and accelerates you less. However, it does accelerate you. If you were to start, say, halfway between the center of the Earth and the surface, you would feel exactly the same acceleration at that point as you would if you had started at the surface of the Earth. So Ms vS's statement is wrong: your motion (speed) doesn't come into this at all.

Addendum added July 5, 2000 Initially, I wondered if this was an editing error. If you remove the word ``you'' from the above statement, you get

Although gravity would get weaker as you approach the center, your motion would keep accelerating.

which is correct! So maybe an overzealous editor added the ``you'', making the explanation incorrect. Or it may have been in there all along. It looks like it was really her error all along; in her column from March 7, 1999 Ms. vS has an interesting addendum to this topic. She says:

Also, my focus on gravity caused some readers to 
misunderstand my statement, [...] Although gravity
would get weaker as you approach the center, your
motion would help to keep you accelerating.... 
A few readers thought I was suggesting that motion
causes acceleration. It does not. Instead I was 
referring to the relationship between force and motion
as expressed by Isaac newton..."