Monday, September 8, 2008

Welcome, XKCD readers!

To all of you that got here from the forums at fabulous xkcd.com: welcome. And yet, many of you make me despair, as I did on that fateful January 31 day. Even as you read the main argument below, the debate is raging anew at the forum of your favorite webcomic. Read the latest comments for the explanation and the debate even continues there! Some of you still don't get it. I die a little inside every time I hear someone say that the plane will not take off, and cite their experience as a "commercial airline pilot", or boast their thousands of hours of flight time, as if some desperate appeal to authority will make up for a complete lack of understanding of basic physics. So I implore you, please - if you read the entire article below, and still think the plane doesn't take off, read it again. And if you still think it doesn't take off, read it again, only make sure you read it this time. We all know you didn't pay attention in physics class, but hopefully you at least stayed awake for a tiny portion of reading comprehension. Read it again, and again, and read the comments, and the thousands of other websites that explain why the plane takes off, and if you still think it doesn't, then please stop clogging the tubes of the internets with mindless drivel. Try one last video that might convince you otherwise instead. Happy flying.

282 comments:

1 – 200 of 282   Newer›   Newest»
Anonymous said...

It is utterly disgusting that all the truth on this marvellous subject is not yet written on Wikipedia...

Anonymous said...

I found the video explained it best.

But seriously, I had never heard of this debate before, and after reading this post don't understand how it can be debated (assuming your physics are correct, as the skateboard example seems to demonstrate).

Christopher said...

Although I agree that with any practical treadmill, the plane takes off. There are so many subtleties in this question.

You say that there are two correct answers:
No, the plane can't take off
and
Yes, the plane can take off

I disagree. The question says This conveyor has a control system that tracks the plane speed and tunes the speed of the conveyor to be exactly the same (but in the opposite direction)
so if the plane is moving forward then the control system has failed.

Therefore the two possible answers are not fly / no-fly they are:
No, the plane won't take off
and
The question is invalid since you can't build such a treadmill.

But this is a thought experiment, we can build whatever treadmill we like. Therefore as soon as the plane starts to accelerate relative to the ground our treadmill spins up to crazy speeds and all sorts of effects will provide some extra friction and stop the plane.
At the practical end of the scale we'll see effects like the grease in the wheel bearings contributing to the friction (in moving through fluids faster is definitely harder); obviously at some stage the wheels will melt which will probably stop the take-off. At the extreme end of the scale eventually relativity will mean that the kinetic energy of the wheels will give them enough mass to either exceed the plane's take-off weight or just to increase the friction within the bearing.

Anonymous said...

Christopher if the treadmill goes in the same (but oposit speed) of the aircraft the when the plane moves forward slowly the the treadmill moves backwards slowly. not very fast in order to keep the plane stationary since the plane is not moving very fast forward. This means that the wheels only need to remain intact at twice the take off speed.
(take off speed forward + take off speed backwards)

Christopher said...
This comment has been removed by the author.
Christopher said...

Well that's only true if you assume that "the plane speed" means its speed relative to the ground not the treadmill.

That's an odd assumption though. If you assume that then you don't need a plane, you can just walk off the runway. Walk forward at 2mph, treadmill moves back at 1mph and you're going forward relative to the ground at 1mph.

I suppose it's still a possible interpretation of the question but it means that you don't make use of any of the special properties of planes. Presumably the original question was supposed to demonstrate some sort of aeronautical point so I don't think this interpretation can have been the original intention.

Also, it's not consistent with the normal use of a treadmill. If I said that I "went to a gym, got on a treadmill and adjusted the speed so that it was the same as my running (but in the opposite direction)", would you assume I ran in to those TVs that they have in gyms playing MTV at half my normal running speed or would you assume I stayed still relative to the gym floor?

toothlesscomb said...

The point you're ignoring christopher is that the plane's speed has no bearing at all to it's friction on the ground. It's friction remains constant (only increasing by a slight degree due to the friction of the grease and bearings) while it's speed increases. The treadmill cannot transfer enough force to stop the plane from moving relative to the ground, since a plane moves by pushing air, not the ground. If it was a car with wings, then yes, you'd be correct. But as it uses a jet engine (or a propeller, or what have you) it doesn't increase it's surface friction to increase speed.

Southwest said...

I hope you don't mind that one of the first few posts in the respective xkcd forum thread was me posting a link here. It helped me understand the thought experiment, and I figured that the benefits of having more people well-versed in this problem would outweigh the possibly overwhelming traffic.

That said, thanks for the site! I found it incredibly helpful in wrapping my mind around this.

Tom said...

All you 'yes fly'er's, get of your little skateboards and learn about relativity. If you put your skateboard on your treadmills, yes they will stay relatively stationary without any external force. Now give your skateboards a push. Yes they will over come the treadmill moving backwards and move forward, and I guarantee you that the speed of the skateboard moving forward will be equal to the speed of it's wheels minus the speed of the treadmill. This is called relativity. Also note that whilst we exist in a world with gravity, the forward movement of the skateboard will always be dependent on the speed of the wheels.

Do you see what has happened here, the external force (not applied through the wheels, like an airplane's thrusters) increased the speed of the wheels for the skateboard to move forwards relative to the treadmill.

So, if there is a command in the question stating that the treadmill will travel at the same speed as the wheels of the plane, by some complex control system, the force imparted to increase the speed of the plane, and hence it's wheels, will cause the treadmill to increase speed also, but in the opposite direction.

So, assuming no external lift, every time the pilot increases his thrust, he will increase the speed of his wheels, which will increase the speed of the treadmill, which will increase the speed of the wheels, which will increase the speed of the treadmill, which will increase the speed of the wheels ... do you see where this is going?

Anonymous said...

Tom:
No, the wheels of a plane are free wheels. A free wheel isn't going to speed up the treadmill.

Well, minor correction: if the axle is completely frictionless it won't have an effect on the speed of the treadmill. Since there is a small amount of friction in the bearings the plane's engines will speed up the treadmill a little bit, but the additional speed will not increase when the pilot increases to full throttle.

Dan said...

Now give your skateboards a push. Yes they will over come the treadmill moving backwards and move forward, and I guarantee you that the speed of the skateboard moving forward will be equal to the speed of it's wheels minus the speed of the treadmill.

I think you're getting your causality wrong. You're assuming that:

Speed of Skateboard = Speed of Wheels - Speed of Treadmill

In fact:

Speed of Wheels = Speed of Treadmill + Speed of Skateboard

When you push the skateboard, it starts to go forwards, which means the wheels spin faster. If you then increase the speed of the treadmill, you will not succeed in reducing the speed of the skateboard, only in making the wheels spin *even faster*.

Thinking about it, this is basically a restating of Xeno's paradox, in which Achilles and a Tortoise are replaced with an Aeroplane and a Treadmill. At low speeds, the 'plane could theoretically match velocity with the treadmill such that there was no movement (scenario 2 in the original post) but there comes a point where the treadmill simply ceases to be a factor.

Kenneth said...

Wow. Rickrolled.

Clever.

Seriously, I've never actually been successfully Rickrolled. I normally expect it. But this was classic.

Also. I fully expected that link to be to the Mythbusters video on the topic.

Jake said...

Thanks for turning me from 'no-flyer' to 'flyer', I never truly understood the problem. Even the mythbusters did a pretty crappy job explaining the problem. Thanks for laying it all our for me.

Christopher said...

@comb

"only increasing by a slight degree"

There's the crucial point. Over certain ranges friction is _almost_ independent of speed. Almost but not quite and that fraction is enough. There's no limit to the speed that the treadmill can run so if there is even the slightest real life deviation from the ideal constant friction wheels then eventually the treadmill will be able to either balance the thrust of the engines or (probably more likely) destroy the plane.

Josh said...

Christopher, draw a Free Body Diagram of the wheel. What causes the wheels to spin? It is not the thrust of the airplane, which acts on the center of gravity of the wheel, it is the static friction from the treadmill, f=ks*N, which gives it a torque, causing rotation. The friction force doesn't increase with speed, because it is a based on two constants, the coefficient of static friction and the weight of the airplane. So, as long as the thrust from the engines is greater than the static friction of the wheel on the surface, it will always take off.

faultyschematics said...

the whole point of this blog was to get you idiots to stop debating it. you could at least have enough class to debate it elsewhere.

Christopher said...

Josh,
You're right, for an ideal wheel.
However, what speed will our wheels and treadmill be rotating at once the plane starts to move? It should be rotating at it's maximum speed to try to keep the plane still but if it's an ideal treadmill it doesn't have a maximum speed.

So the question really is whether you consider that the treadmill is ideal or the plane.
I took the treadmill to be ideal because we're told in the question that the treadmill CAN track the plane's speed. The three rewordings make it even clearer by referring to the treadmill as "very powerful".

I think that most of the problem here comes from the fact this site claims to be the definitive analysis but then goes on to give a single yes answer. The definitive analysis would surely but a matrix of assumptions with an answer for each. Unfortunately the original question is so hazy that it will be quite a large matrix.

Anonymous said...

Hey, just wanted to say Randall just blatantly ripped you off:

http://blag.xkcd.com/

You might want to send him a nice email about this.

Original Sin said...

I just got rickrolled.

/golfclap

Gregory said...

......
If I didn't laugh my ass off every time I was Rickrolled, I'd come to your house and cut you.

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Anonymous said...

If the turbines are spinning, won't they push the air across the wings regardless of the planes movement?

sbunny8 said...

It seems to me that the original problem was this: "If you have a plane that needs a 2000ft runway to get up to 70mph before it can take off, and you didn't have enough room to build a 2000ft runway, could you build a 40ft treadmill, crank it up to 70mph, and take off straight up?". I think we can all agree unanimously the answer to that question was NO IT WOULDN'T WORK.

Then some unknown smart-aleck said "oh yeah? What if the treadmill was 2000ft long? You could rev up the plane's engine and move forward along the treadmill and yes you'd be able to take off at the end of it, even if the treadmill operator cranked up the speed of the treadmill way faster than 70mph." And guess what, the smart-aleck is right. To which I say SO FREAKING WHAT?? Why on earth would anyone want to build a 2000ft treadmill when they could just build a 2000ft runway? You've defeated the whole point of the original puzzle.

Anonymous said...

the definition on the XKCD page DIFFERS from the one in this blog.

This blog discusses "The speed of the belt is matched to the airplane's speed"

XKCD's version says "conveyor belt is designed to exactly match the speed of the wheels".

It's important, because the answers are actually different

Optimus said...

I wasn't much aware of the problem and when I tried in the past to read it and read some answers I never quite got it. Initially I would think no-fly but that's because I didn't have an idea of airplanes and the wheel not being mechanical. And of course remembering it's the air thrust that moves it and the wheel just for rolling nicely.

But the bicycle/skateboard explanation made it for me. It's everyday things that we are more familiar, like when you have a grinding wheel or a motor, and on the other hand you put a sellotape ring on your pen, keep it stationary so that the grinding wheel friction rotate it backwards, no matter how faster you didn't need to excert more counter force. It reminds me of little games we played, I remember how simple the force it has to be to just counteract the friction.

Anyway, was a great read, fortunately I never had an opinion on this problem when I came here, so maybe I wasn't biased and that's why I read and tried to understand it. I had an easier go with Monty Hall problem, but this one I always avoided it or never knew what the fuzz was about.

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Ramuan Alami Cerebrovascular
Komposisi Anti-addiction Capsule
Rahasia Pria Kuat Diatas Ranjang
Cara Ampuh Melangsingkan Badan
Khasiat Eye Care Softgel
Cara Diet Ampuh dan Alami
Terapi Pengobatan Radang Sendi dengan Mujarab
Pengobatan Alami Pengapuran Tulang Tumit Aman
Green World Lecithin Softgel
Rahasia Melangsingkan Badan
Obat Herbal Melangsingkan Badan

Obat Baik said...

Solusi Pria Hebat Gagah Perkasa saat Bercinta
Bahaya Luka Lambung
Ramuan Alami Impotensi Paling Ampuh
Penyebab Retina Mata Robek
Komposisi Eye Care Softgel
Apakah Operasi Kanker Rahim Berbahaya ?
Testimoni Vig Power Capsule
Bahaya Kanker Payudara Mematikan
Obat Pelangsing Berbahaya Menurut BPOM
Tanaman Herbal untuk Disfungsi Seksual
Pengobatan Radang Sendi Terampuh
Obat Kuat Pria di Apotik
Nama Obat Ejakulasi Dini Paling Ampuh
Racikan Obat Tradisional Asam Urat Manjur
Cara Ampuh Merawat Sendi agar Tidak Mudah Nyeri

Obat Baik said...

Rahasia Melangsingkan Badan
Obat Herbal Melangsingkan Badan
Pantangan Makanan Penderita Hati Berlemak
Cara Mengobati Mata Berkabut Alami
Efek Samping Slimming Capsule
Pir Penurun Berat Badan
Bahaya Operasi Kista
Keajaiban Buah Tomat Mengobati Kanker Ptostat
Obat Pelangsing Badan dalam 7 Hari
Cara Ampuh Melangsingkan Badan
Khasiat Eye Care Softgel
Cara Diet Ampuh dan Alami
Pengobatan Alami Radang Sendi Tumit Kaki Terbaik
Green World Chitosan Capsule
Racikan Obat Tradisional Stroke

Obat Baik said...

Rahasia Melangsingkan Badan
Obat Herbal Melangsingkan Badan
Pantangan Makanan Penderita Hati Berlemak
Cara Mengobati Mata Berkabut Alami
Efek Samping Slimming Capsule
Pir Penurun Berat Badan
Bahaya Operasi Kista
Keajaiban Buah Tomat Mengobati Kanker Ptostat
Obat Pelangsing Badan dalam 7 Hari
Cara Ampuh Melangsingkan Badan
Khasiat Eye Care Softgel
Cara Diet Ampuh dan Alami
Pengobatan Alami Radang Sendi Tumit Kaki Terbaik
Green World Chitosan Capsule
Racikan Obat Tradisional Stroke

Data Obat said...

Bahaya Paru Paru Kotor karena Asap Rokok
Efek Samping Slimming Capsule
Cara Ampuh Tahan Lama Malam Pertama
Pir Penurun Berat Badan
Gejala dan Penyebab Susah BAB
Khasiat Eye Care Softgel
Jenis penyakit pada Tiroid
Cara Ampuh Melangsingkan Badan
Pisang Buah Ajaib Pelangsing Tubuh
Rahasia Melangsingkan Badan
Obat Herbal Melangsingkan Badan
Resep Alami Herbal untuk Infeksi Pencernaan
Cara Diet Ampuh dan Alami
Apa Nama Obat Ambeien yang Mujarab
Pengobatan Alami Radang Sendi Panggul dengan Cepat

Obat Multikhasiat said...

Komposisi Eye Care Softgel
Cara Menghilangkan Lendir Paru Paru Alami
Obat Kuat Pria di Apotik
Apa itu Eksoftalmus
Tanaman Herbal untuk Disfungsi Seksual
Penyebab dan Pengobatan Kanker Otak Mujarab
Nama Obat Ejakulasi Dini Paling Ampuh
Solusi Memiliki Tubuh Singset paling Ampuh
Ramuan Alami Impotensi Paling Ampuh
Testimoni Vig Power Capsule
Bahaya Ginjal Bocor Mematikan
Green World Kidney Care Capsule For Man
Eye Care Softgel Harga
Ramuan Herbal Alami untuk Mengobati Sinusitis
Spesialis Obat Tradisional Wasir yang Manjur

Bahaya Tumor Otak said...

Pir Penurun Berat Badan
Nama Obat Ejakulasi Dini di Apotik
Cara Ampuh Melangsingkan Badan
Rahasia Melangsingkan Badan
Racikan Obat Herbal Lambung Bengkak
Efek Samping Slimming Capsule
Tanaman Obat Herbal Diabetes Melitus
Obat Herbal Melangsingkan Badan
Apakah Kanker Prostat Berbahaya
Khasiat Eye Care Softgel
Cara Diet Ampuh dan Alami
Bahaya Tumor Otak
Green World Koloklin Capsule
Pengobatan Alami Rabun Senja paling Manjur
Pantangan Makanan Penderita Infeksi Lambung Terbaru

Obat Ampuh said...

Penyebab dan Pengobatan Lambung Berdarah Alami
Penyebab Mata Menjadi Buta
Komposisi Eye Care Softgel
Makanan Sehat Penderita Penyakit Hati
Pengobatan Ampuh Ejakulasi Dini secara Alami
Nama Obat Disfungsi Seksual paling Ampuh
Bahaya Impotensi di Usia Muda
Obat Kuat Pria di atas Ranjang
Khasiat Vig Power Capsule
Cara Alami Memutihkan Kulit secara Menyeluruh dan Merata
Glucoberry dan Glucocoa
Obat Glucoberry dan Glucocoa Nutrisi Pemutih Herbal
Vig Power Capsule Ajaib
Green World Diasis Care Capsule
Makanan Sehat Baik dikonsumsi Penderita Asam Urat

Obat Alami said...

Gejala, Penyebab dan Pengobatan Mata Minus
Pengobatan Kanker Kolon Tanpa Operasi
Penyebab Kelenjar Tiroid Membengkak
Ramuan Obat Herbal Infeksi Paru Paru
Tips melangsingkan Badan 1 Minggu
Obat Alami Pelangsing Tanpa BKO
Cara Ampuh Menurunkan Berat Badan Alami
Obat Tradisional Meningkatkan IQ anak secara Alami
Terapi Pengobatan Osteoporosis Ampuh
Pantangan Makanan Penderita Infeksi Pencernaan
Efek Samping Slimming Capsule
Solusi Melangsingkan Badan secara Cepat
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Terapi Pengobatan Infeksi Lambung Alami

Obat Alami said...

Penyebab Mata Plus pada Anak
Racikan Obat Herbal Hepatitis Ampuh
Terapi Pengobatan Pasca Operasi Ampuh
Obat Herbal Glucoberry dan Glucocoa Nutrisi Pemutih Kulit
Obat Tradisional Glucoberry dan Glucocoa Nutrisi Pemutih Kulit
Pengobatan Ampuh Ejakulasi Dini secara Alami
Nama Obat Disfungsi Seksual paling Ampuh
Manfaat Eye Care Softgel
Bahaya Impotensi di Usia Muda
Pengobatan Alami Meningkatkan Daya Tahan Tubuh Ampuh
Green World Super CoQ-10 Capsule
Obat Tradisional Maag Akut paling Manjur
Obat Kuat Pria di atas Ranjang
Khasiat Vig Power Capsule
Ramuan Obat Herbal Kanker Hati Efektif

Obat Alami said...

Penyebab Mata Silinder pada Anak
Pengobatan Ampuh Ejakulasi Dini secara Alami
Nama Obat Disfungsi Seksual paling Ampuh
Manfaat Eye Care Softgel
Ramuan Obat Herbal Darah Tinggi
Bahaya Impotensi di Usia Muda
Bahaya Operasi Kanker Payudara
Penyebab Hati Sirosis
Obat Kuat Pria di atas Ranjang
Khasiat Vig Power Capsule
Nama Obat Radang Paru Paru Alami
Pantangan Makanan Bagi Penderita Ambeien Terbaru
Makanan Pantangan Bagi Penderita Sinusitis
Pengobatan Alami untuk Memicu Kecerdasan Anak
Green World Breast Care Capsule

Obat Alami said...

Gejala dan Penyebab Mata Belekan
Tips melangsingkan Badan 1 Minggu
Obat Alami Pelangsing Tanpa BKO
Bahaya Diabetes Melitus Mematikan
Cara Ampuh Menurunkan Berat Badan Alami
Efek Samping Operasi Kanker Prostat
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Ciri-ciri Radang Lambung pada orang Dewasa
Efek Samping Slimming Capsule
Solusi Melangsingkan Badan secara Cepat
Obat Tradisional Kencing Manis paling Mujarab
Tanaman Herbal Alami untuk Mengobati Kanker Serviks
Obat Tradisional untuk Memenuhi Nutrisi Mata
Green World TriFlex Capsule

Pepaya Lancarkan Susah Buang Air Besar said...

nice and pefect

Akun Semangat said...
This comment has been removed by the author.
Akun Semangat said...

Tetes Obat Mata Bengkak
Cara Ampuh Menurunkan Berat Badan Alami
Ciri-Ciri Penyakit Hepatitis Membahayakan
Macam-Macam Penyakit Tiroid
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Alkohol Penyebab Utama Infeksi Hati
Obat Herbal Mata Bintitan Alami
Tips melangsingkan Badan 1 Minggu
Obat Alami Pelangsing Tanpa BKO
Penyebab dan Pengobatan Nyeri Lutut
Efek Negatif Operasi Kanker Otak
Efek Samping Slimming Capsule
Solusi Melangsingkan Badan secara Cepat
Produk Green World Indonesia Asli

Biaya Operasi Kista Rahim said...

nice gan

Airi Obat said...

Bahaya Retina Berdarah
Tips melangsingkan Badan 1 Minggu
Obat Alami Pelangsing Tanpa BKO
Bahaya Darah Tinggi penyebab Stroke
Efek Samping Slimming Capsule
Solusi Melangsingkan Badan secara Cepat
Ramuan Herbal Kanker Kulit Ampuh
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Penyebab Lambung Bocor pada Anak
Cara Ampuh Menurunkan Berat Badan Alami
Pengobatan Alami Selulitis Orbitalis Ampuh
Obat Tradisional Perut Kembung yang Ampuh
Waspadai Gejala Kanker Hati dan Penyebabnya
Ramuan Tradisional Alami untuk Radang Paru Paru

Bahaya Usus Buntu Tanpa Operasi said...

nice and perfect

Tanaman Alami Mampu Memenuhi Nutrisi Mata said...

keren abis

Bahaya Obat Pelangsing Mengandung BKO said...

sukses selalu buat kita semua

Terapi Pengobatan Sakit Pinggang Terampuh said...

benar benar unik

Biaya Operasi Batu Ginjal 2016 said...

tolong menolong dalam kebaikan..

Gejala dan Penyebab Ginjal Bengkak said...

Tetap optimis kawan..

Airi Obat said...

Penyebab Mata Gatal Karena Alergi
Solusi Melangsingkan Badan secara Cepat
Resep Obat Alami Pengencer Darah Ampuh
Cara Ampuh Menurunkan Berat Badan Alami
Penyebab Adanya Benjolan di Kelopak Mata Berbahaya
Bahaya Kanker Kulit Tanpa Pengobatan
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Bahaya Kanker Darah Tanpa Cuci Darah
Virus Penyebab Radang Hati
Tips melangsingkan Badan 1 Minggu
Terapi Pengobatan Nyeri Sendi paling Ampuh
Obat Alami Pelangsing Tanpa BKO
Efek Samping Slimming Capsule
Obat Tradisional untuk Meningkatkan Nafsu Makan Pada Anak

Cara Menuntaskan Kanker Prostat Tanpa Operasi said...

succes for all

Obat Multikhasiat said...

Obat Alami Pelangsing Tanpa BKO
Efek Samping Slimming Capsule
Penyebab Mata Perih dan Gatal
Cara Ampuh Menurunkan Berat Badan Alami
Pantangan Makanan Penderita Leukimia
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Tanaman Obat Gagal Hati Ampuh
Tips melangsingkan Badan 1 Minggu
Tips Turunkan Berat Badan paling Mujarab
Distributor Vig Power Capsule Bandung
Cara Ampuh Mencegah Disfungsi Seksual Sejak Dini
Solusi Melangsingkan Badan secara Cepat
Obat Tradisional Paru Paru Basah paling Ampuh
Pantangan Makanan untuk Penderita Radang Paru Paru

Bahaya Stroke Ringan yang Disepelekan said...

Happy Weekend all my friends..

Pelangsing wsc biolo said...

happy info nice sahre

obat pelangsing badan cepat said...

salam semangat maksih beritaya

obat penurun berat badan cepat said...

obat penurun berat badan cepat

obat penurun berat badan cepat said...

obat penurun berat badan cepat

OBAT PENURUN BERAT BADAN ALAMI said...

OBAT PENURUN BERAT BADAN ALAMI

Obat penurun berat badan aman said...

Obat penurun berat badan aman

Obat pelangsing badan cepat dan ampuh said...

Obat pelangsing badan cepat dan ampuh

Obat diet cepat dan aman said...

Obat diet cepat dan aman

OBAT PELANGSING WSC BIOLO ORIGINAL said...

OBAT PELANGSING WSC BIOLO ORIGINAL

OBAT PELANGSING AMPUH said...

OBAT PELANGSING AMPUH

Kapsul Pelangsing Biolo said...

Kapsul Pelangsing Biolo

Obat Pelangsing Biolo Woo Tekh said...

Obat Pelangsing Biolo Woo Tekh

Obat Pelangsing Badan Alami said...

Obat Pelangsing Badan Alami

Obat Diet Alami Cepat said...

Obat Diet Alami Cepat

Obat Herbal Penurun Kolesterol Tinggi said...

Obat Herbal Penurun Kolesterol Tinggi

Obat Herbal Penurun Kolesterol Jamkho said...

Obat Herbal Penurun Kolesterol Jamkho

Jamkho Herbal Kolesterol Dan Asam Urat said...

Jamkho Herbal Kolesterol Dan Asam Urat

Obat Pelangsing Slimming Capsule Green World said...

Obat Pelangsing Slimming Capsule Green World

Krim Pemutih Wajah Cepat Aman said...

Krim Pemutih Wajah Cepat Aman

Cream Pemutih Wajah Aman Dan Bagus said...

Cream Pemutih Wajah Aman Dan Bagus

Cream Pemutih Aman said...

Cream Pemutih Aman

Toko Obat Herbal Online Terpercaya said...

Toko Obat Herbal Online Terpercaya

Data Obat said...

Solusi Melangsingkan Badan secara Cepat
Bahaya Mata Merah Karena Kelilipan
Efek Samping Slimming Capsule
Biaya Operasi Kanker Kulit 2016
Cara Ampuh Menurunkan Berat Badan Alami
Pantangan Makanan Penderita Cacar Air
Tips melangsingkan Badan 1 Minggu
MultiVitamin pelangsing Badan paling Ampuh
Jual Obat Kuat Vig Power Capsule Bali
Obat Alami Pelangsing Tanpa BKO
Pengobatan Penis Peluh paling Manjur
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Bahaya penyakit Gondok yang Tidak Diobati
Obat Tradisional untuk Memicu Penambah Tinggi Badan

Obat Alami said...

Bahaya penyakit Gondok yang Tidak Diobati
Obat Tradisional untuk Memicu Penambah Tinggi Badan
Biaya Kemoterapi Kanker Darah
Obat Kuat Pria di atas Ranjang
Bahaya Impotensi di Usia Muda
Bahaya Campak Jerman Pada Ibu Hamil
Slimmng Capsule Solusi Langsingkan Badan Terampuh
Manfaat Eye Care Softgel
Nama Obat Disfungsi Seksual paling Ampuh
Pusat Penjualan Vig Power Capsule Indonesia
Khasiat Vig Power Capsule
Pengobatan Ampuh Ejakulasi Dini secara Alami
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Penyebab dan Pengobatan Penis Loyo

DISTRIBUTOR OBAT HERBAL said...

MAKASIH SANGAT berguna

toko obat herbal online said...

sangat bermanfaat

toko obat herbal online terpercaya said...

salamsemangat makasih

obat penurun berat badan aman said...

makasih informasinya

Obat Pelangsing Badan Alami Wsc Biolo said...


Obat Pelangsing Badan Alami Wsc Biolo

Obat Pelangsing Biolo Woo Tekh said...

Obat Pelangsing Biolo Woo Tekh

Obat Pelangsing Tubuh Yang Ampuh said...

Obat Pelangsing Tubuh Yang Ampuh

OBAT penurun berat badan ampuh said...

salam semangat makasigh ritanya

OBAT pelangsing wsc biolo said...

salam sehat makasih beritanya

Obat pelangsing badan cepat dan ampuh wsc biolo said...

Obat pelangsing badan cepat dan ampuh wsc biolo

Chitosan Capsule Woo Tekh said...

Chitosan Capsule Woo Tekh

Obat pelangsing badan cepat said...

Obat pelangsing badan cepat

obat pelangsing yang aman said...

obat pelangsing yang aman

Pelangsing wsc biolo woo tekh said...

Pelangsing wsc biolo woo tekh

OBAT TRADISIONAL PELANGSING AMAN said...

OBAT TRADISIONAL PELANGSING AMAN

biolo world slimming capsule said...

biolo world slimming capsule

Wsc Biolo Murah Dan Asli said...

Wsc Biolo Murah Dan Asli

pelangsing tradisional cepat aman said...

pelangsing tradisional cepat aman

Obat tradisional herbal alami said...

Obat tradisional herbal alami

Obat Herbal Penurun Kolesterol Tinggi Jamkho said...

Obat Herbal Penurun Kolesterol Tinggi Jamkho

Obat Kolesterol Dan Asam Urat Tinggi said...

Obat Kolesterol Dan Asam Urat Tinggi

Jamkho Obat Kolesterol Manjur said...

Jamkho Obat Kolesterol Manjur

Bitter Melon Softgel Woo Tekh said...

Bitter Melon Softgel Woo Tekh

Obat Herbal Penurun Kolesterol Jamkho said...

Obat Herbal Penurun Kolesterol Jamkho

JAMKHO OBAT KOLESTEROL said...

JAMKHO OBAT KOLESTEROL

Jamkho Herbal Kolesterol Dan Asam Urat said...

Jamkho Herbal Kolesterol Dan Asam Urat

Krim Pemutih Wajah Cepat Aman said...

Krim Pemutih Wajah Cepat Aman

Cream Pemutih Wajah Aman Dan Bagus said...

Cream Pemutih Wajah Aman Dan Bagus

Krim Ampuh Penghilang Flek Hitam said...

Krim Ampuh Penghilang Flek Hitam

Krim Ampuh Penghilang Flek Hitam said...

Krim Ampuh Penghilang Flek Hitam

Cara Menghilangkan Bintik Hitam Di Wajah said...

Cara Menghilangkan Bintik Hitam Di Wajah

Obat Pelangsing Slimming Capsule Green World said...

Obat Pelangsing Slimming Capsule Green World

Chitosan Capsule Woo Tekh said...

Chitosan Capsule Woo Tekh

Chitosan Capsule Dari Woo Tekh said...

Chitosan Capsule Dari Woo Tekh

Pantangan Makanan Penderita Kanker Otak said...

terima kasih beritanya.. semoga semakin sukses..

Cara Menghilangkan Jerawat Super Cepat said...

Cara Menghilangkan Jerawat Super Cepat

OBAT DIET said...

MAKASIH

Obat Alami said...

Nama Obat Disfungsi Seksual paling Ampuh
Tanaman Obat Leukimia Ampuh
Persamaan Rubella dan Campak Jerman
Manfaat Eye Care Softgel
Slimming Capsule Penurun Berat Badan Ampuh
Khasiat Vig Power Capsule
Pengobatan Ampuh Ejakulasi Dini secara Alami
Jual Vig Power Capsule Apotik Palembang
Obat Kuat Pria di atas Ranjang
Bahaya Impotensi di Usia Muda
Jamu Kuat Pria diatas Ranjang
Waspadai Gejala Infeksi Paru Paru Karena Mudah Menular
Obat Tradisional TBC Tulang paling Ampuh
Obat Tradisional Abses Paru paling Ampuh
Waspadai Gejala dan Penyebab penyakit Asma

Bahaya Penyumbatan Sirkulasi Darah ke Otak said...

sukses gan

Obat Alami said...

Obat Kuat Pria di atas Ranjang
Bahaya Impotensi di Usia Muda
Bahaya Ulkus Kornea pada Mata
Khasiat Vig Power Capsule
Pengobatan Ampuh Ejakulasi Dini secara Alami
Obat Tradisional Kekentalan Darah Ampuh
Manfaat Eye Care Softgel
Penyebab Vitalitas Pria Rendah
Tanaman Alami Ampuh Pelangsing Badan
Pusat Penjualan Vig Power Capsule Manado
Nama Obat Disfungsi Seksual paling Ampuh
Racikan Obat Mati Rasa pada Kelamin Pria
Pengobatan Alami untuk Meningkatkan Konsentrasi Belajar
Obat Tradisional Hepatitis B paling Ampuh
Obat Tradisional untuk Melindungi Lensa Mata

Racikan Obat Herbal Jantung Bengkak Alami said...

yakin dan optimis..

OBAT PELANGSING GREEN WORLD said...

SELAMAT MALAM salam semanat

green world slimming capsule gwsc said...

salam sehat makasih

wsc biolo obat pelangsing badan said...

makasih sangat atas informasi yang bermanfaat ini

Gejala dan Penyebab Mata Lelah said...

amazing very nice

PELANGSING wsc biolo said...

salam semangat makasih beritanya

Yashodara Cream Pemutih Wajah Aman said...

Yashodara Cream Pemutih Wajah Aman

OBAT DIET cepat aman said...

OBAT DIET cepat aman

obat diet ampuh said...

obat diet ampuh

Pelangsing wsc biolo woo tekh said...

Pelangsing wsc biolo woo tekh

biolo obat pelangsing yang aman said...

biolo obat pelangsing yang aman

pelangsing badan cepat dan ampuh wsc biolo said...

pelangsing badan cepat dan ampuh wsc biolo

Obat diet cepat dan aman wsc biolo said...

Obat diet cepat dan aman wsc biolo

OBAT PELANGSING WSC BIOLO ORIGINAL said...

OBAT PELANGSING WSC BIOLO ORIGINAL

toko obat herbal online terpercaya said...

toko obat herbal online terpercaya

obat penurun berat badan ampuh said...

obat penurun berat badan ampuh

KAPSUL pelangsing biolo said...

KAPSUL pelangsing biolo

obat pelangsing wsc biolo said...

obat pelangsing wsc biolo

Obat Pelangsing Badan Alami Wsc Biolo said...

Obat Pelangsing Badan Alami Wsc Biolo

Obat Diet Alami Cepat said...

Obat Diet Alami Cepat

Distributor Obat Herbal Berkualitas said...

Distributor Obat Herbal Berkualitas

Apotik Obat Herbal Online said...

Apotik Obat Herbal Online

Obat Herbal Penurun Kolesterol Tinggi Jamkho said...

Obat Herbal Penurun Kolesterol Tinggi Jamkho

Obat Kolesterol Dan Asam Urat Tinggi said...

Obat Kolesterol Dan Asam Urat Tinggi

Jamkho Obat Kolesterol Manjur said...

Jamkho Obat Kolesterol Manjur

Maderik Obat Herbal Asam Urat said...

Maderik Obat Herbal Asam Urat

Obat Herbal Penurun Kolesterol Jamkho said...

Obat Herbal Penurun Kolesterol Jamkho

JAMKHO OBAT KOLESTEROL said...

JAMKHO OBAT KOLESTEROL

Jamkho Herbal Kolesterol Dan Asam Urat said...

Jamkho Herbal Kolesterol Dan Asam Urat

Krim Pemutih Wajah Cepat Aman said...

Krim Pemutih Wajah Cepat Aman

Cream Pemutih Wajah Aman Dan Bagus said...

Cream Pemutih Wajah Aman Dan Bagus

Cara Mengencangkan Kulit Wajah Yang Kendur said...

Cara Mengencangkan Kulit Wajah Yang Kendur

Cream Memutihkan Wajah said...

Cream Memutihkan Wajah

Obat Pelangsing Slimming Capsule Green World said...

Obat Pelangsing Slimming Capsule Green World

Obat Alami said...

Obat Alami Pelangsing Tanpa BKO
Efek Samping Kemoterapi Kanker Darah
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Gejala dan Penyebab Disfungsi Seksual
Ramuan Pengurus Badan paling Ampuh
Tips melangsingkan Badan 1 Minggu
Efek Samping Slimming Capsule
Jual Obat Kuat Vig Power Capsule di Apotik Bengkulu
Cara Ampuh Menurunkan Berat Badan Alami
Rahasia Ampuh Menambah Gairah Seksual Pria
Solusi Melangsingkan Badan secara Cepat
Obat Tradisional Menambah Berat Badan Alami
Obat Tradisional Hepatitis C paling Manjur
Obat Tradisional Glaukoma paling Manjur

Pantangan Makanan Penderita Migrain said...

semoga semakin sukses semua aamiin..

Obat Alami said...

Efek Samping Slimming Capsule
Ramuan Kemoterapi Kanker Tulang Alami
Cara Ampuh Menurunkan Berat Badan Alami
Makanan Alami Penuntas Mata Minus
Solusi Melangsingkan Badan secara Cepat
Suplemen Pelangsing paling Jitu dan Efektif
Tips melangsingkan Badan 1 Minggu
Distributor Obat Vig Power Capsule Apotik Padang
Kegunaan Eye Care Softgel
Resep Diet Ampuh dan Aman
Cara Ampuh Mengatasi Sulit Orgasme Pria
Obat Alami Pelangsing Tanpa BKO
Obat Tradisional Penghilang Sendawa
Manfaat Kecubung untuk Meringankan Asma
Obat Tradisional Penggemuk Badan Alami paling Ampuh

Pantangan Makanan Penderita Gagal Jantung said...

can certainly remain optimistic

Obat Alami said...

Efek Samping Bahaya Kemoterapi Kanker Darah
Pengobatan Ampuh Ejakulasi Dini secara Alami
Gejala dan Faktor Risiko Glaukoma
Tanaman Ajaib Pelangsing Badan Tercepat
Obat Kuat Pria di atas Ranjang
Bahaya Impotensi di Usia Muda
Nama Obat Disfungsi Seksual paling Ampuh
Distributor Penjualan Vig Power Capsule Banyuwangi
Efek Samping Konsumsi Vig Power Capsule
Manfaat Eye Care Softgel
Rahasia Memperbesar Alat Vital Pria
Khasiat Vig Power Capsule
Manfaat Tanaman Ciplukan untuk Mengobati Gondok
Vig Power Capsule Harga
Obat Tradisional Urat Syaraf Kejepit paling Ampuh

Pengobatan Alami untuk Mengatasi Obesitas said...

nice and perfecrt amazing

Ramuan Menuntaskan Mata Sakit paling Ampuh said...

keren banget artikelnya gan..

Obat Alami said...

Pantangan Makanan Penderita Kanker Tulang succes Obat Kuat Pria di atas Ranjang succes Bahaya Impotensi di Usia Muda succes Gejala dan Penyebab Luka Usus succes Nama Obat Disfungsi Seksual paling Ampuh succes Cara Diet Sehat Terampuh Tanpa Efek succes Pengobatan Ampuh Ejakulasi Dini secara Alami succes Jual Vig Power Capsule Apotik Bogor succes Efek Samping Konsumsi Vig Power Capsule succes Manfaat Eye Care Softgel succes Jamu Pria Kuat Super Mujarab succes Khasiat Vig Power Capsule succes Obat Tradisional Meningkatkan Kekebalan Tubuh Anak succes Obat Tradisional Emfisema paling Ampuh succes Apakah penyakit Radang Paru Paru Menular succes

Ramuan Peramping Badan Alami Aman said...

berusaha dan berdo'a.. Tetap semangat !!!

Rebecca said...

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pelangsing badan biolo wsc said...

ksalam semangat terimakasioh banyap;

Obat pelangsing badan cepat said...

Obat pelangsing badan cepat

Jamu pelangsing tradisional cepat aman said...

Jamu pelangsing tradisional cepat aman

Pelangsing biolo world slimming capsule said...

Pelangsing biolo world slimming capsule

OBAT TRADISIONAL PELANGSING AMAN CEPAT JITU said...

OBAT TRADISIONAL PELANGSING AMAN CEPAT JITU

Pelangsing wsc biolo woo tekh said...

Pelangsing wsc biolo woo tekh

Wsc biolo obat pelangsing yang aman said...

Wsc biolo obat pelangsing yang aman

Obat pelangsing badan cepat dan ampuh wsc biolo said...

Obat pelangsing badan cepat dan ampuh wsc biolo

Obat diet cepat dan aman wsc biolo said...

Obat diet cepat dan aman wsc biolo

OBAT PELANGSING WSC BIOLO ORIGINAL said...

OBAT PELANGSING WSC BIOLO ORIGINAL

OBAT PELANGSING AMPUH WSC BIOLO said...

OBAT PELANGSING AMPUH WSC BIOLO

Kapsul Pelangsing Biolo said...

Kapsul Pelangsing Biolo

Obat Pelangsing Biolo Woo Tekh said...

Obat Pelangsing Biolo Woo Tekh

Obat Pelangsing Badan Alami Wsc Biolo said...

Obat Pelangsing Badan Alami Wsc Biolo

Obat Diet Alami Cepat said...

Obat Diet Alami Cepat

Obat Pelangsing Slimming Capsule Green World said...

Obat Pelangsing Slimming Capsule Green World

Yashodara Cream Pemutih Wajah Aman said...

Yashodara Cream Pemutih Wajah Aman

Fair n Pink Serum Lotion Pemutih Badan said...


Fair n Pink Serum Lotion Pemutih Badan

Jamkho Herbal Kolesterol Dan Asam Urat said...

Jamkho Herbal Kolesterol Dan Asam Urat

Obat Herbal Penurun Kolesterol Jamkho said...

Obat Herbal Penurun Kolesterol Jamkho

Maderik Obat Herbal Asam Urat said...

Maderik Obat Herbal Asam Urat

Jamkho Obat Kolesterol Manjur said...

Jamkho Obat Kolesterol Manjur

Obat Kolesterol Dan Asam Urat Tinggi said...

Obat Kolesterol Dan Asam Urat Tinggi

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