How to mathematically calculate a fall through the Earth
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- เผยแพร่เมื่อ 21 ก.ย. 2024
- Watch the full integration! Worth it even just for the Simple Harmonic Motion which drops out at the end. • Integrating a fall thr...
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. • Race the ISS through t...
Python code that I used is here: www.dropbox.co...
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. • How Long To Fall Throu...
CORRECTIONS:
- 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: / mattparker
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MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
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Nerdy maths toys: mathsgear.co.uk/
"Your fragile face scraping against the rock surface will cause friction, which is frankly inconvenient for the maths."
Norimori
@@jayveeascano4570 jayvee Ascaño
@@collin5577 collin
@@vwlz8637 vwlz
@@yoyoyonono Aaron Thapa Ch. - Games, Music and Stuff
Repeatedly falling through the hole, emerging at the other end of the sphere and plummeting once again is technically orbiting (mathematically).
if you watched the full video, he actually says this
“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?
My instinct is that the assumption that the density of the Earth is a constant changes the answer less than other assumptions but that is just a guess. If a geologist can give me an agreed density profile of the Earth I'll re-do the calculations!
Cody'sLab or you would just get stuck in the center.
CraftQueen, only if you include air resistance. Otherwise that greater mass will accelerate you downwards faster so you are going at a higher speed when you go past the middle and the greater decceleration will be unable to stop you.
But, since the Earth is symetrical, wouldn't the differences cancel out so that the overall density is all that matters, since we're talking about going through the entire Earth through the center? In which case, if you're not going through the center, then you would have to recalculate that.
Nope.
As an extreme example, consider all the mass located at a single point at the centre.
In that case the acceleration due to gravity will be the standard GM/r^2.
That applies all the way until you reach the centre, with the rate of acceleration increasing the closer you get.
Conversely, with uniform density you have M=4/3*pi*rho*r^3.
Thus a=4/3*pi*rho*r^3/r^3=4/3*pi*rho*r.
This means acceleration would be greatest at the surface.
Or a more extreme counterexample, the entire mass is a spherical shell at the surface.
Then in passing through this shell you accelerate at roughly 9.8 m/s^2 then don't accelerate at all until you come out the other side.
So for these 2 extremes, one has the smallest acceleration on the surface and greatest at the centre, the other has the greatest on the surface and least under it.
Walking through Fremantle today and my son grabs my sleeve - "Dad, Dad, that's the guy who does those cool TH-cam videos!" I learned two things today:
- The world is smaller then you think.
- My son is a bigger nerd than I realised.
+Turn it off! Yes, that was me! I'm from Perth originally; was out for a day with my siblings. Be sure to say hello next time! Say hi to your son for me.
Imagine being the boy that comes home from school and his dad goes "Matt Parker told me to tell you, what's up"
LOL! There's much hope for at least one kid; Nurture that nerd-ness, please. (and it might help to teach him some cool fighting/defensive style, bullies being what they are ... ignorant and belligerent, in my experience.) I was the 3rd-grader who discovered how NEAT slide rules were and other nerdy things.
@@omikronweapon Heck! Imagine being the PARENT.
The re-capping of the sharpie at 4:32 was done so professionally.
18:20 Imagine that, you fell all the way through the earth and came out alive on the other side only to get obliterated by a tennis ball going at 28,000km/h that you yourself threw. That's karma for daring to drill through our beloved earth I tell you.
Complicated suicide doesn't get too much simpler than that.
69 likes
0:13
"a apple"
Yoshimaster96smwc Lol
Parker grammar.
"Certainly it can core a apple."
-Chef of The Future
Well, he is a Maths teacher, not an English teacher... ;)
Apple is just an instance of a Earth.
2:07 Until you said "Oh grow up" I never noticed. THANKS MATT.
"assuming there is no air resistance"
me, a highschool student: "air resistance? What air resistance?"
me: "what about magma resistance?"
@@renedekker9806 Just gave you a thumbs up on that remark. Although imho there's no need to take that into consideration. Because by definition the hole goes all the way through the earth. Ergo you won't touch any magma. What is actually not mentioned in the set-up definition is the heat. I assume you'd evaporate long before you reach the earth's core. But still you won't have to fear that because you'd be already dead before the temperature becomes really uncomfortable. I'm just not sure if you die from your blood vessels bursting or from cardiac arrest, both due to the acceleration... On a more serious note: What about the earth's rotation? Would that affect the fall in a way that you'd hit the wall of the hole? I think not, but I'm not sure about that. I suppose it's the same as when jumping straight upwards and landing on the exact same spot although the earth has been spinning underneath you.
@@ellesko Dam, the point of a cheeky reply is to make it short.. Go tired of reading at the second line!
@@ellesko actually, since gravity isn't a real force, you don't feel acceleration until you encounter resistance. So, if you fall, and there is no air to resist you, you would feel weightless until you hit the ground.
is what you never assume.
"Oh grow up!"
What, were you saying it was silly to think we'd just bounce back and forth?
(Thirty entire seconds pass)
Wait...
I was in math mode, so I didn't get it. I had go into teen mode
@@BLSXful Which explains why teens are so useless at math... except for the nerds, of course.
He should have used a peach for that joke.
@@jrbleau Tbh, you're right. It took me failing a math class to kick it into high gear.
Actually, I didn't spot that one as I was too busy pointing out the fact that you're exerting a pull on the earth as well, which, hmm... on second thought, does it matter? Would you end up stationary at the centre of the earth, or would both of you wobble around your barycentre?
Hm, on third thought I think I answered that one myself (it's the latter, so they cancel out. Right?)
Anyway, my point is, maths is no joking matter.
2:13 how did I not see that? What's happened to me? Did I accidentally remove my mind from the gutter?
I wouldn't want to alarm you, but it's often even worst than we like to think at first glance. You might be succumbing hopelessly into disgusting seemliness, where only grief and sorrow awaits you. Quick, go refresh your mind with delightful Sandler's comedies before it's too late. (In fact just watch them all in one go, can't be too safe.)
Sandler movies? I want my mind in the gutter, not the sewer :P
I'm afraid the sewers are your last and only hope...
+Elf Friend He said "Grow up."
saw it just before he said "grow up". then I said "sorry mom"
Matt, what on EARTH are you doing (@9:28)?!
You're using a dot as both a decimal and thousands separator!!!
Henrik Wilson He’s offending both sides of the debate at the same time, just to keep everything balanced.
Yea, or shall I say what the fall are you doing
When Matt pointed out 17K MPH as the max velocity, I thought "That's the velocity of the Space Shuttle in orbit" (when we had one) The he explained the rest. VERY COOL STUFF.
I actually had this problem as an exercise demonstrating harmonic oscillation in my high school physics textbook. Good times.
It is actually more fun with air resistence. You will have damping that willl give you some decaying oscillation that will likely end you sitting susupended in the middle of the earth.
I knew the answer would be 42.
I was about to comment the exact same thing, before looking in the comments section for the same answer.
Life the universe and everything
@@alexanderschmidt6787 I like the way of your thinking
I had to like this. I was number 42.
ninjapancake you are a god
This question came up as a free response question on my physics final and I had no idea how to do the math but I remembered this video, wrote 42 minutes and got full credit 😎
You got full credit on a math question on a test without showing any work? That's... concerning.
@@IceMetalPunk well to be fair, he said it was a free question on a physics final so...
@@QPUNeptune He said it was a "free response" question, as opposed to a multiple choice, true/false, or fill in the blank question. One where you write freeform to answer it. That doesn't mean the question is free...
@@IceMetalPunk welp, my fault, i misread that, thanks for pointing that out
That doesn't reflect well on the college.
Isn't this assuming that the Earth's mass is even distributed? Most of the Earth's mass comes from the very center.
in the description, he linked to a minutephysics video that takes it into account
If the more dense part is at the center this would still work as its still symetric. But if the denser bit is on a side it wouldn't work.
Yes and no. Technically you're right, but everything is statistically scattered extremely even from the centre, that you don't need to do that. You really only need the complete mass and the distance.
+71866B
Actually because the mass is non evenly distributed it takes less time to fall through the earth.
The centre is more dense.
Its kind of similar to how better solutions to the Brachistochrone problem arent a straight line. Matt is assuming wrong in this case and is overestimating the time.
Just imagine that you have a small 50kg ball. The centre of mass is exactly in the middle of that ball. If you now put that into a hollow light ball that has a bigger space, the centre of mass is still in the middle. That's why you don't need those 2 mass seperatively if the centre of mass is exactly in the middle. This is 100% with the earth. All elements on the earth have different distances to the centre and has different mass, but all those have the same centre, that's why you don't need to seperate things.
Wait... in the 'London to New York' part of the video, wouldn't you need a non linear path/tunnel? Gravity would not pull you straight from London to NY but instead would pull you along the side of the tunnel towards the lopsided mass side... So how did we manage to get the same answer for time? Did we take a parabolic route sling-shot-ing around the core of earth similar to a gravity assist in space?
This is the one part of the video I really didn't understand very well. He said it's take the same amount of time no matter where you drilled to. What if I drilled a hole to my neighbors yard? I doubt it'd take me 42min to walk through that tunnel.
@@damionwhitehead1165 I think this is assuming you drill a hole to at least the center of the earth... still a little iffy in my opinion
It does seem dubious, a hole that doesn't pass through the center of the planet isn't going in a downwards direction and you'd just fall in to the slanted wall of the tunnel.
I have been watching these math channels for years and they are so amazing. I started watching these videos when i was a freshman in high school taking algebra 1, now i am in 12 grade AP Calculus and AP Physics. The math has gone from complete gibberish to things I understand. Its so cool. So to all of the math channels, thanks for being a part of my life and thanks for being one of the few intelligent corners left on youtube. ❤️ love you guys
this had to involve the apple, the iconic symbol of gravity.
+James Humphreys That did please me.
"This is not going to be exact... I'm using a food product."
"Will cause friction which is, frankly, inconvenient for the maths." It's also inconvenient for the face that's scraping along the rock.
As though 42 minutes wasn't a good enough connection, "There is an art to flying, or rather a knack. The knack lies in learning how to throw yourself at the ground and miss. ... Clearly, it is this second part, the missing, that presents the difficulties." The hitchhikers guide to the galaxy is the best trilogy ever and if you don't think so I'll fight you
I love how he says "oh, grow up!" before it even occurred to me what was going on. Thanks for putting your ideas into my mind Matt. ;)
Why didn't you cut the onion the other way? Then the layers are much more concentric and you can peel them easier.
Yeah, I face-palmed when it happened.
He wanted to use Parker Spheres.
It was the Parker Cut.
It was a Parker Square of a demostration.
Cooking habits are almost impossible to break
The Answer to Life, the Universe and Everything, plus 10.5 sec!
Is that a coincidence??
John Violidakis and how 42/4 is 10.5...and 4 has 4 letters! it's all connected!
Betcha when the magratheans built earth it was exactly 42m - there were just too many meteorite collisions happening since then :P
@Eli
Soo, your telling me that it's a coincidence that coincidences look like coincidences because otherwise they wouldn't be a coincidence?
Wouldn't that be a coincidence?
p.s. I was trying to make sense so I kept re-reading my comment before I posted. This made me forget the meaning of coincidence and now I feel like I just made up the word so I'm just gonna post this before I go even more crazy.
Exactly what I was thinking!
John Violidakis
It is not.
For falling through a hole not on a central axis, you'd end up scraping against the walls I assume
He is incorrect because the majority of the mass would be on one side. Therefore the force would not up in the same direction as the direction that you are falling.
At 10:50, my first thought was "Gee, if only we had a planet with an earth-like chemical makeup and a diameter of at least 299792.458 * 60 * 42.18 Kilometers!"
You can also substitute the density, rho, for the mass of the earth over the volume of the earth (we already assumed constant density) to get ρ=3M/4πr^2, and then the 4/3s cancel and you get sqrt(π^2 r^2/GM)
"oh, grow up"
you got me 😂
I probably laughed more than I should have at 2:12 !
This has been one of the videos that gave me the most laughs! Well done for a math lecture!
From apples to onions to whiteboard and chalkboard too!
Your first kinematics equation is written as v = ut +1/2at^2 . That v should be the displacement (s) rather than the final velocity (v).
The video begins: Stick man falling through onions and apples...
The video ends: SO, we just integrated multiple derived equations of motion to derive harmonic oscillators that describe the gravitational influence on objects falling in and around a sphere...
3:13 that seems a bit counterintuitive. I've always thought of it as if I'm in a basement, I'm closer to the center of the earth, so that gravity has more of a pull on me. But I guess his logic makes sense, because if you continue this until you reach the center, you are "essentially" weightless. But if you go higher than the surface, your weight will also decrease, so the surface is where you weigh the most?
No, the peak is a couple thousand miles down. He was wrong at that point.
@@bryanreed742 No, at least if the density is the same everywhere (which it isn't).
When you go up above the surface, the mass that pulls you down stays the same but your distance increases -> less Force.
When you go below the surface, the mass that pulls you down gets cancelled by mass that is above you and pulls you up. You are also MUCH closer to the mass pulling you up - the Molecules in the concrete above you "count" a lot more than the Molecules down in Australia (or Sweden, if you happen to live on the Southern Hemisphere). In the end, while the distance to that centerpoint decreases, the Mass of that thing is also decreasing. The Distance gets squared, but Mass is dependent on Volume, and that gets cubed, so the Force in total decreases as well.
"a apple"
"oh, grow up!"
How can i "fall" not through the middle of the earth? im always pulled to there? i would have to walk?
or like using a sledge
without friction
+haskia jimmi Yes a sledge.
+
Same way as planets or comets don't have to fall into the sun. Have a little sideways speed and you'll miss the center.
Great timing! We just discussed this topic in my Classical Mechanics course.
My train of thought as Matt is demonstrating the ideal motion through the planet given a perfect system: "I need to gif this."
Matt: "Oh, grow up!"
that feeling when you're so early that HD is not available yet but it doesn't matter anyway because your internet is really slow, yeah, it's painful.
I have HD available, so it apparently becomes available fast
+Lavaloid the Human You were so early you had the first comment!
How slow is your internet? Mine is 0.65 mbps, max.
I wish there was a love button because this is an absolutely awesome video. I loved it to bits.
You know when you've grown up is when you realize the dirty joke after he says "grow up"
Hahaha, that's what I like about these kinds of jokes. When it works for both people who have grown up, and those who haven't grown up
Scared me when I heard him say that because he caught me at the height of my shame
In fact, there are no grown ups, just people with different complex minds, some are more childish and some are used to think they are grown up, that's all :D
I guess I'm not grown up because before he even put the apple together I was anticipating it.
@@Hextator Me too!
In October 1965, at the age of 17, I had an interview at King's College London, to enter their BSc Special Physics course the following year. The interviewer asked me this exact question. I DID manage to work out that it was simple harmonic motion, but no way to calculate the actual period in the heat of the interview. He had to tell me the match to the sea-level satellite period. But I still got the place on the course ...
So glad this exists. There's an episode of Batman Beyond where a villain misusing technology that allows him to phase through matter winds up falling through the Earth. The episode ends with Bruce saying he'd stop falling at the core, but when I asked the physics teacher at my school he explained exactly this (though in super condensed form).
JUST SO EVERYBODY KNOWS, BECAUSE APPARENTLY I'M THE ONLY ONE:
Today is international Power of 2 Day!!!
At 1:02, the date and time will read 2^0:2^1 2^2/2^3/2^4
Very cool! Of course, here in America that happened back in April :p
I wish I'd known!
Gawsome technically that's the time and date
@@michaels4340 in Britain they use the weird format: dd/mm/yy, so they have a completely different date for it
@@gamingwithgabe6156 dd/mm/yy is the only logical solution. XD
yy-mm-dd or mm-dd-yy is just people on drugs and a big ego justifing their ideas and believes without thinking a split second about the advantages of other ideas. like using empiric measuring system.. XD degree fahrenheit, lol.. when ever it sounds plausible to use a measuring scale that got fit to the coldest day in danzig and the body temperature of a sick inventor, then you are in the states (he had high temperature that day, lol)
address and date go from fine detail to coarse detail aswell:
First Name Last Name
Street
City
State
Country
@@gamingwithgabe6156 Yes, I'm aware!
8:00 Lost me at 'density of the Earth doesn't change', especially after all that earlier mention of the core.
This is a glaring inaccuracy that should be mentioned in the video! The earth's density increases with depth, and in particular the core (being mostly iron) is much more dense than the rest, being rock.
Gravity increases (at first) as you go down into the earth from the surface! (Mostly because you get closer to the core.) However, the end result is still the same, which I imagine standupmaths would say is really cool.
@@jlittlenz While it's still symmetric and you'll still reach the other side with a net velocity of zero, you'll arrive there sooner due to the different acceleration profile so I'd say the end result isn't still the same.
@@WJS774correct me if i am wrong, the amount by which the acceleration is happening due to increasing density while moving towards the center is same as the amount of deceleration due to decreasing density once u move past the center, since the force is decreasing. like if density is "g" then
f is proportional to g but since g is not a constant and is changing
we can say
f is proportional to g - dg/dr +dg/dr which cancels out if the density of the starting and the ending point is same.
the question i want to ask is why should the time decrease
where r is the instantaneous position reference to the center of the earth
@@alidurrani4645 Imagine the extreme case where the mass of the earth is in fact a point mass. In that case, gravity _increases_ the closer you get to the centre. The greater acceleration means greater velocity and greater velocity means less time. Obviously the earth is not in fact a point mass, so the gravity does not increase _all_ the way to the centre, but it _does_ increase until you reach the core.
I'm not sure if I missed it during the explanation, but wouldn't it be more of a circular horizontal slice than just the sphere around the center that determines your gravitational pull force?
Like, some of the spheres "above" you as you fall have a portion of the sphere below you as well, so instead of an onion wouldn't it be like a line perpendicular to the falling direction?
Because (In your scenario in which the earth is all the same density, which it isn't actually) it doesn't matter that it's a part of the outer sphere(s), it's still pulling on you with the same mass as before and that pull would have a vertical and horizontal portion of its pull vector, but the horizontal portions would get counteracted by the same portion from the opposite direction, so just the vertical part would remain
Do the numbers work out to the same results with this line of reasoning?
Falling through the center of the earth is a special case where the horizontal components, using your terms (I might call them the y and z axes), effectively cancel each other out at all points, leaving only the vertical component to affect your acceleration. The two parts of the onion is just a useful visual to simplify the problem into two parts (the sphere below you and the "shell" above you). The shell of the onion can be further split (see the cut he made) into two parts which have equal and opposite effects on you in ALL directions.
I have been a fan for years and was pleasantly surprised today when I came across you on a tv science show talking about rainbows! What a delight. You are awesome.
+Vanessa W I do that sometimes. But TH-cam is my home.
Your enthusiasm is totally catching!
4:04 You made a mistake here. You could have cut the onion accross the "equator" if you will and you would have got more concentric circles.
Another Parker Square: drawing on the inside of an apple with a Sharpie.
2:07 I like that a lot...................
This takes me back. Did this during introductory theoretical physics back in the day. :)
well, you take 42 minutes (and a bunch of seconds) to fall from one spot to any other spot on earth (the math going on is absolutely amazing btw.) and you take this time for any planet no matter which size but with the same density as earth, may this is the thing that they wanted to point out, when they said “forty-two is the answer of all questions“ in the movie “the Hichhiker's guide to the galaxy“!! I'm absolutely amazed!! mind's blown!!
"It's very hard to draw on an onion."
Ah, yes. The kind of quality information I came here to learn.
"Oh grow up!"
Hey Matt, I just finished this, as always fantastic, video and the saw the sponsoring at the end. Which I think is great to keep your channel up and running. But then I was wondering whether there was another way to support you? Since I'm from Germany, I was only able to buy a digital copy of your "festival" and couldn't show up in person - btw, love it! Something like flattr or patreon or at least a PayPal account to donate something. I love your channel and your videos and I understand that you're doing this in your spare time, so give us a chance to support you even more! Lovely greetings from Aachen, Germany.
Completely agree. Would also support him on patreon :P
+Benedikt Geiben Thanks! I'm now convinced I should set up a Patreon. Would be nice to have the stability to keep making videos.
Great, really looking forward to it!
I don't know if in the 1,619 comments this was addressed, but if you take a simple pendulum of a certain length, and you take a hula hoop of a diameter equal to that of the length of the simple pendulum (ignoring the lack of isochronism, of course), your Periods would be identical! I noticed that during this excellent discussion. Great work on the video!!! Thanks a million!
I love it!
I have watched many of your videos.
I really liked "How to mathematically calculate a fall through the Earth".
The only issue I have, is that "Rho" would be changing significantly as you fell.
Because of chemical fractionation, when the Earth was molten,
all the denser materials became concentrated in the Inner and Outer Core.
Call it "Metal Hurlant", if you will: Iron, Nickel and other "Heavy Metals".
The Mantle is much less dense due to all of the silicates, and the crust is the rarest of all.
And it is not just a trivial difference either...
The Core is estimated to be a factor of 3 to 4 times denser than the outer layers.
It seems like this would lead to a significantly different result.
"Journey to the centre of the onion," the far less exciting sequel.
"The reason I'm using [the onion]... it's made of concentric spheres."
Yeah, onions have layers. Like ogres.
24:06 So, if the Earth fell through the sun, surviving intact, it would reach the opposite side of its normal orbit in the same amount of time? whoa
jpaugh64 only if the Earth orbited at the surface level of the Sun
The time would change based on the sun's density.
I really hope it doesn't...just sayin'
Funnily enough the way Matt drew the "grazing" satelite it's actually higher than the orbit of the ISS. The orbital period of the ISS is also 90-93 minutes, so with half that being 45-46.5 minutes, that's a negligible difference from the 42 minutes you would get at the surface.
Can you explain the side hole more in depth? my understanding of this problem up to now was that your path must pass straight through the center of the planet so all side gravitational forces cancel out.
So, it would take me 42 mins to fall to me neighbors house? What would that even be like?
Mitchell if your neighbour lives on the other side of the world, I don't think he is your neighbour
If there was no friction and a direct path then yes
Well, technically, you couldn't. Gravity would pull you to the center of the earth; you can't fall perpendicular to gravitational forces. However you could theoretically set up an oscillation between the two points *using* gravity, which would make that time correct. To answer your question, it would be a nearly 45 minute slide along a frictionless sidewalk.
Mitchell think frictionless skateboard over a flat area. The force is not quite perpendicular
This was used as an example of theoretical forms of transportation. If we could build a train over a completely frictionless track in a completely straight vacuum tunnel between any 2 points on the earth's surface (exact same distance to the earth's center of mass), the train would take those 42ish minutes to do the trip using gravity alone. If the two points happen to be a few meters apart, the distance is very short but the acceleration would also be minuscule.
Not even kidding my eyes started watering when he cut the onion 😮
This was great but I still don't understand how you put a hole in that apple so fast.
“28,474 kph, that is pretty fast” is the understatement of the century
We finally have the question to the ultimate answer of life, the universe and everything. 42 = how long to orbit or fall through a planet? Simply stunning.
by making that hole, would nt it decrease the mass of earth
Not unless the mined material was being lauched off the planet...
Would, when falling from London to new York, the gravity of the center of the earth be great enough to pull you along the wall of the tunnel causing friction?
I think you would not be falling at all. You could actually just walk through that hole ;D
All of these measurements are assuming that you're falling directly through the center of the tube with no friction.
It would also require some sort of friction-less rail system, yes. I thought of this too.
Not the gravity of the center, but the coriolis effect.
Caleb Brinyark If the walls are lined with Teflon and you were covered in butter you'll be fine sliding along the side
So I have two big problems with this video.
The first is when you mentioned the situation in which one falls through a tunnel that doesn't go through the center of the Earth. If one were to jump into such a hole/tunnel, gravity will still pull them directly towards the center of mass of the Earth, so they wouldn't fall perfectly centered through the hole, and would eventually hit the wall of the tunnel, making it impossible to fall through such a hole/tunnel without an external force to keep them centered within that hole. I guess if you can create a device that fights against your off-centered acceleration towards the wall of the tunnel in order to keep you centered, making it only affect that and nothing else, then you'd fall through the off-centered tunnel in the same amount of time, but if it's a system where only gravitational force is applied, then it's only possible to fall through the center of mass.
The other problem has to deal with time dilation when gravity changes, which affects how you would calculate the time it takes to cross the planet depending on which point-of-view you're calculating from, but that's on a whole new level of complication, so I don't fault you for not taking that into account.
this was so entretaining. i saw years ago minutephysics video about the same subject. but the math perspective on this one is a great bonus.
Had an interesting thought, since the time taken for an object to fall through a planet/body is independent of the size of said body, say we have a planet with uniform density that is more than 7.59 x10^11 metres in diameter, does this mean that the falling object will exceed the speed of light? 7.59x10^11m/2530.5s = 299940723.2 m/s > c
Assuming of course that said planet's density is equal to that of the Earth.
00:13 Matt: "a apple"
Me thinking: "Were my kindergarten books wrong?!"
So basically what you're saying at 4:04 is that onions are like ogres?
someBODY once told me the world was a gummy so I took a bite out of a tree.
ONIONS. HAVE. LAYERS.
Shrek is love. Shrek is life.
@@shorterneilisbored7078 it tasted kind of funny so spit it at a bunny that started cursing at me
11:50
What is bugging me the most is that in reality, the density of the sphere below you is not constant, it's a function of x!
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.
(Just noticed that sqrt(miles/kilometers) almost exactly = 54/42. Hm.)
9:45 just thinking he should mention it doesn't matter if your going through the center or not, then saw video was'nt even half over, good on ya Matt
Matt: "Assuming there's no air resistance"
Me: High-School Physics all over again
I didn't even think about it untill he said: oh grow up.
I feel like you glossed over the point that the gravity above you is cancelled out by the gravity of the shell outside your altitude below you. Is that something to do with the way the inverse square law for gravity falling off over distance cancels out the volume of a hollowed sphere?
That's also my assumption.
Matt parker! thanks for the videos! always looking forward to them when i get home, after a long day of work, only to turn into mush what ever chunks of brain i have left. I appreciate it.
14:20 I would think that splitting it into 1m chunks would actually be _more_ effort than the integration even using a computer.
"is someone cutting onions in here?"
hahahaha
The answer to everything is 42!
but the density of the earth does change. I'm pretty sure that the molten / solid iron core (we're not sure what state it's in under such great temprature and pressure) has a higher density than that of the mantle.
Ok, but we're already talking about a ridiculous situation, so we can probably assume uniform density
Actually that bit at 20:40 made me actually understand it for the first time. Of course it has to be the same time. Because if you throw a ball sidewise and drop a ball at the same time, they will fall the same distance and land on the ground at the same time.
And your Satellite is basically just a ball that starts with sideways movement and falls straight down just like if you fall through earth - just that it also gets pulled to the side aswell.
This prompted me to _finally_ look up the shell theorem. I've been wondering how to prove that literally for years, but i've never found it before because... well, maybe i just didn't look hard enough.
2:13 was specifically at me :(
Thomas P i was waiting for something like this and what like no he can’t be that professional
Not like any other youtubers have done this ........
As far I know they didn't calculate time it takes and speed.
How many people have you seen play the same game?
I wonder how many of these actually used integration ;)
Nobody owns the facts
how many textbooks covering the exact same material exist? how many of them are published by the same companies? how many of them are written by the same people? not getting to some bit of information first on youtube is not a reason not to do a video.
2:00 how to bαng tutorial
the orbit at 0 and the time taken to fall through are linked despite what you say! Orbits are basically just falling (with style) except you miss the planet in the process. From the side your corenaut and the orbit will be identical.
You know what's even more amazing? It will ALWAYS take the SAME amount of time no matter from where the hole starts, and wherever it ends (assuming ideal conditions). It could literally be 2m away from the start point, as long as that end point is also on the circumference of the Earth.
EDIT: Nevermind got too excited. He did say it later on.
I'm disappointed. That's not even remotely a coincidence. The ball's motion along the "y" axis is the same. It just also happens to be moving along the x axis as well.
+notanimposter Yep, that's the video in a nutshell.
I have an idea for the material and/or cosmology analysis. When you are on the planet of unknown mass, make a tunnel through the core, fill it with the vacuum (unless it's already) and measure the travel time. Congratulations - now you know the density of that planet and thus, assuming you know the radius, its mass.
And the period of a simple pendulum of length R as well! Pendulum, orbit and drop all the same! when I discovered this in high school I was amazed!
it makes sense that the time is the same orbit vs falling through because an orbit is just falling through a planet while dodging it
With all due respect yo your more recent videos (which I thoroughly enjoy BTW), this is my favorite video on your channel!