How to mathematically calculate a fall through the Earth

in Coincidence by

Huge thanks to the Great Courses Plus for sponsoring this video. Click here to check out the free trial:

Watch the full integration! Worth it even just for the Simple Harmonic Motion which drops out at the end.

Try the ISS challenge! Will you get there before the after the International Space Station orbits half-way around the planet? Even if you don’t try to work it out, you can have a guess at the outcome.

Python code that I used is here:

My values:

R = 6,371,000 m (radius of the Earth)
G = 6.674 × 10^-11 m^3 kg^-1 s^-2 (gravitational constant)
M = 5.972 × 10^24 kg (mass of Earth)

Yes, people have since pointed out a similar topic was covered in a Minute Physics video. They skip over the “constant stuff” and how the mathematics is derived, but it’s a great look at how the density changes within the Earth and how that impacts the travel time. Well worth checking out as well.

– I accidentally wrote “v” instead “s” on the board in “s = ut + ½at^2”. First spotted by Joel Low.
– Around 9:30 I used a dot for both 1,000’s and decimal point. The first should be a comma. Spotted by Mezgrman.
– I normally play pretty fast and loose with centripetal force vs centrifugal effect; I think arguing about the difference is not useful so I often use them as synonyms. In this case, as Wayne Ernst politely pointed out, I should have said “centripetal force” not “centrifugal force”. And they’re right.

Discuss on reddit:

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MATT PARKER: Stand-up Mathematician
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  1. The density of the earth is not constant, most of the mass is in the core, the mantle is a lot less dense, and the crust is the least dense (well, the atmosphere is less dense but that's already above most of us)

  2. So, if you are a little lighter underground, and you are also lighter when you go very very far away from earth, does this mean you are heaviest at the surface of a planet?

  3. And what if this Earth-like planet is so big that we sould travel faster than the speed of light to get to the other side in 42m 10.5s?

  4. 17:50 but that coincidence isnt really a surprise is it?

    the satelite is also falling and you can calculate the falling time by chancing the density constant in you equation because the mass doesnt change but the volume does?
    (i hope you understand what i mean dear reader xd)

  5. I was thinking about how to make a perpetual motion machine from this and he asked me to grow up, now I'm sad

  6. doesn't the acceleration also change depending on the acceleration due to gravity caused by the body of the object you're going through? and then wouldn't that change the time required if the body had a large circumference but low acceleration caused by gravity?

  7. But… If you dig a tunnel through the side of the planet, will you really fall through? Wouldn't just have to walk on the side of the tunnel closer to the planet's core? Because you know… that's what we normally do with our tunnels.

  8. My calculus teacher doesn't like the word cancel, she calls it the "c" word in our class and doesn't like it when people use it, because if 3π/π cancels out to 3 and ln(e) cancels out to 1 and (1/2) * 2 cancels out to 1, you are using the same word for completely different operations. The π's don't cancel out, they divide out, the ln(e) doesn't cancel out, they inverse out, and the (1/2) * 2 doesn't cancel out, they multiply out. She would rather people say what the actual operation is instead of saying that they always just "cancel." Either divide out, inverse out, et cetera, just not cancel.

  9. Right. Love your stuff! Glad the Great Courses is helping out. Must say its now been about 45 years, and its always interesting what you recall after a long span of time. There were 8 of us in that particular class during university days, and we each arrived at an identical result, which was memorable in itself. Fifty four minutes. That is the number we got. Offhand I can't say where our assumptions, methods or reference values may have deviated from yours, but a 30% variance can't be dismissed as mere experimental error. I believe that if I search diligently enough I will turn up the old notebook with those calculations, so perhaps an additional comment later.

  10. So that means if I have a sphere of water at 4 degree celcius, and i put a metal ball on one end of the sphere, ignoring the friction, it will get to the other end in 1 hr 39 mins and 1.6 secs. That's fast!

  11. 0:13 wait, what just happened? How? I refuse to believe this, you magician, burn the sorcerer! You had an apple with a solid center, threw it in the air, it's center was hollowed out. You, good sir, I do believe have broken the rules.

  12. Your math is completely wrong and I'm not even talking about the centrifugal forces involved. Even pretending Earth's mass is evenly distributed and no friction occurs, still way off.

  13. Rho does change, because it is defined as the density of the mass in the sphere below you, which becomes more hotter and dense as you get closer to the center of mass

  14. К стенке притянет где-нибудь внутри, даже если не раздавит чудовищным давлением. А чё если пробурить вокруг земли туннель, откачать воздух и запустить тело подобно тому, как спутники по орбите?

  15. “Earths density does not change” well that just triggered all the geologists. The center of the earth is much more dense than the exterior. In fact the core is a mars scized chunk of iron! Now if you were to just fall through the iron bit the time to fall through would less than the 42 minutes. This means that going through the core would take LESS TIME! Right? Or am I missing something?

  16. Anyone else want to clean that chalk board more? I'd take that job. Clean it really well before each video. Just to have my name in the credits 😀

  17. Really sharp knives prevent much of the tears, reason is the sharp edge forces out less juice thus less tears

  18. Assuming that a = acceleration, v = velocity, s = displacement, t = time, u = initial velocity:
    That first motion equation should be "s = ut + a(t^2)/2", not "v = ut + a(t^2)/2".

  19. I don't ge it: if you "fall" from London to New York, you do not fall towards the center of earth, that is the earth's center of gravity. So you are not pulled towards the center of your tunnel but towards its "downsided" edge. Why is the speed/time the same, nonetheless?

  20. I don't like pi… Why isn't it a nice number in our mathematics? Is it the only way to calculate circles?

  21. Since we're neglecting friction, ect., I would like to point out this is not technically always correct. If the body was large enough (and we'll assume that somehow it doesn't collapse into a blackhole and you aren't destroyed by gravitational tidal forces) then you would eventually accelerate to near the speed of light. As a result, it would take longer than the aforementioned time in this video to outside observers. I'm not sure it's even a resolved problem on precisely what the person experiencing the constant acceleration would read on his clock. Of course, this is another nonphysical assumption, but it is an interesting exercise in its own right.

  22. nice video lol, we had to do the same calculations in first semester physics as a project but i couldnt really handle already

  23. 2:14 I sat there for a minute trying to figure out why he told me to grow up.. completely missed it the first time

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