Down the rabbit hole
Down, down, down. Would the fall NEVER come to an end! ‘I wonder how many miles I’ve fallen by this time?’ she said aloud. ‘I must be getting somewhere near the centre of the earth. Let me see: that would be four thousand miles down, I think–‘ (for, you see, Alice had learnt several things of this sort in her lessons in the schoolroom, and though this was not a VERY good opportunity for showing off her knowledge, as there was no one to listen to her, still it was good practice to say it over) ’–yes, that’s about the right distance–but then I wonder what Latitude or Longitude I’ve got to?’ (Alice had no idea what Latitude was, or Longitude either, but thought they were nice grand words to say.)
Presently she began again. ‘I wonder if I shall fall right THROUGH the earth! How funny it’ll seem to come out among the people that walk with their heads downward! The Antipathies, I think–‘ (she was rather glad there WAS no one listening, this time, as it didn’t sound at all the right word) ’–but I shall have to ask them what the name of the country is, you know. Please, Ma’am, is this New Zealand or Australia?’ (and she tried to curtsey as she spoke–fancy CURTSEYING as you’re falling through the air! Do you think you could manage it?) `And what an ignorant little girl she’ll think me for asking! No, it’ll never do to ask: perhaps I shall see it written up somewhere.’
If you have read Alice in Wonderland , you may remember this part.
Actually, it can be a very good question -- How long will it take to make Alice fall THROUGH the earth and get to the antipode?
Answer the time in minutes and round to the nearest integer (Since we don't want Alice to be very confused).
Assumptions:
1.The earth is a uniform sphere with the radius .
2.The gravitational acceleration on the surface of the earth is .
3.Neglect the Earth's rotation and the temperature of the centre of the earth. (That is, Alice will not die from evaporation during the fall)
4.Neglect the air resistance . (Although in real world it would matter a lot)
The answer is 42.
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1 solution
Time to fall through the earth is half the time period of SHM Alice will perform, which is π√( g R ). Substituting the values we get the time as 42.217 minutes.