The plane takes off, 0.999... = 1, but what about...

[jagec]

Originally posted by: BrownTown
The plane is moving forward becasue the motion of the wheels and the forward motion are not related assuming that the parts are well oiled and such. In reality there would be a little additional drag due to the spinning wheels, but it would be neglidgible unless the conveyor belt was going reallf really fast.

Actually, there is a way to keep the plane from taking off. But you need either really heavy wheels, or indestructible wheels, because 2 x the energy of all the fuel on the plane will end up in angular momentum :Q

...well, less than that due to inefficiencies and friction, but the point is you CAN keep the plane on the ground if you really try.

Originally posted by: BigPoppa
Easy to baby step someone through this, mix a cup of water and a cup of sugar. Wow its not 2 cups of volume? Proceed to liquids.

Ponder: Can gases be "miscible" in each other or does it not work at that point? Trying to baby step my way through that one, liquid + gas, but not giving me the visual I want. Haven't ever covered like that in any of my chem courses.

Most gases, at most temperatures and pressures that we're used to dealing with, behave as ideal gases. In other words, PV=nRT; any additional moles of gas go directly to volume (at a set temperature and pressure). So they usually do exhibit "conservation of volume" IF you aren't changing T + P.

 

[her209]

Originally posted by: DrPizza

Originally posted by: BrownTown
DrPizza, attempting to point out the fact that mass is lost in a chemical reaction is really pushing it. For all practical purposes mass is considered to be conserved. I guarentee you if I took a chemical engineering class and said i couldn't solve a problem using conservation of mass becasue the chemical reaction would lose mass the professor would fail me. (and probably call me an idiot too)

Is mass conserved in a chemical reaction? No. Is the loss in mass negligible? Yes. The answer to the second question does not change the answer to the first question.

Mass is conserved otherwise the amount of energy released will be much much much more. Just think how much energy is released when a single uranium atom is split.

 

[FrustratedUser]

Originally posted by: JohnCU
If you have 2 liquids (you don't know what they are), 1 mL of each and you add them together, can you be 100% sure that the resulting solution is 2 mL?

Actually.... no you can not be sure of that.

 

[BigPoppa]

Originally posted by: jagec

Originally posted by: BrownTown
The plane is moving forward becasue the motion of the wheels and the forward motion are not related assuming that the parts are well oiled and such. In reality there would be a little additional drag due to the spinning wheels, but it would be neglidgible unless the conveyor belt was going reallf really fast.

Actually, there is a way to keep the plane from taking off. But you need either really heavy wheels, or indestructible wheels, because 2 x the energy of all the fuel on the plane will end up in angular momentum :Q

...well, less than that due to inefficiencies and friction, but the point is you CAN keep the plane on the ground if you really try.

Originally posted by: BigPoppa
Easy to baby step someone through this, mix a cup of water and a cup of sugar. Wow its not 2 cups of volume? Proceed to liquids.

Ponder: Can gases be "miscible" in each other or does it not work at that point? Trying to baby step my way through that one, liquid + gas, but not giving me the visual I want. Haven't ever covered like that in any of my chem courses.

Most gases, at most temperatures and pressures that we're used to dealing with, behave as ideal gases. In other words, PV=nRT; any additional moles of gas go directly to volume (at a set temperature and pressure). So they usually do exhibit "conservation of volume" IF you aren't changing T + P.

Been doing too much organic chem, can't believe I didn't think of PV=nRT. Time to look over inorganic chem, too.

 

[jagec]

Originally posted by: her209

Originally posted by: DrPizza
Is mass conserved in a chemical reaction? No. Is the loss in mass negligible? Yes. The answer to the second question does not change the answer to the first question.

Mass is conserved otherwise the amount of energy released will be much much much more. Just think how much energy is released when a single uranium atom is split.

No, he's right. True, a lot less mass is lost than when a uranium atom is split. But it is well and truly lost. Mass is not perfectly conserved, even in normal chemical reactions.

 

[Tiamat]

Originally posted by: JohnCU
If you have 2 liquids (you don't know what they are), 1 mL of each and you add them together, can you be 100% sure that the resulting solution is 2 mL?

No, you have to consider excess properties and partial molar properties. When two unlike items are mixed, their inter and intra molecular forces interact causing expansion or contraction of the molecules. If you had 1ml of water and 1ml of ethanol you will get something like 1.98mL total.

This is simple thermodynamics people...


Edit: The alcohol does not absorb the water. The intermolecular forces between an ethanol and ethanol molecular are very different from the intermolecular forces between an ethanol and water. Think about this for a little, and you will understand why volume is not conserved, especially in solution.

 

[dxkj]

Wouldnt the energy of the friction on the wheels need to be ~ to the energy being put out by the engines in order to stop of it.


Take it one step further, assume that the wheels are frozen, just two hunks of rubber that dont spin, would the plane take off as long as it could overcome the friction of the wheels.

One step after that, the conveyer belt would have to be moving backwards fast enough that the friction on the immobilized wheels overcame the amount of energy being put out by the engines.

Now think if the conveyer belt was covered in ice, and the tires were replaced by skis.... now you have about the same friction that would happen on a rubber conveyer belt with rubber tires that are connected by a ball bearing axle..

Can you imagine the conveyer belt covered in ice moving backwards fast enough to really affect the plane on skis? Then your answer should be no, it would not stop it from taking off, as the friction is negligible. The conveyer belt would have to be moving at a MUCH MUCH higher speed than the plane in order to create enough friction to slow it down or prevent it from flying.


PS A simpler answer is obvious in the paradox of the question. IF the conveyer belt moves backward at the same speed as the airplane, and the airplane will take off at 200 mph... and the plane is moving forward relative to the non moving ground at 200 mph, then obviously it has the airspeed needed to take off. The wheels are moving at 400 mph, but the friction is low enough for that to barely be an issue

 

[jagec]

Originally posted by: dxkj
Wouldnt the energy of the friction on the wheels need to be ~ to the energy being put out by the engines in order to stop of it.

If the wheels do not slip, a small amount of the energy put into moving the plane forwards has to go towards increasing the angular momentum in the wheels. This is practically negligible, but if we live in ideal-physics-land, where the conveyor is capable of infinite velocity, the wheels do not slip under any circumstances, and the wheels are capable of insanely fast rotational speeds without flying apart, it is possible to stop the plane from taking off. You can even assume zero friction.

But that's a very special set of circumstances. In pretty much all other (more realistic) cases, the plane takes off.

 

[dxkj]

Originally posted by: jagec

Originally posted by: dxkj
Wouldnt the energy of the friction on the wheels need to be ~ to the energy being put out by the engines in order to stop of it.

If the wheels do not slip, a small amount of the energy put into moving the plane forwards has to go towards increasing the angular momentum in the wheels. This is practically negligible, but if we live in ideal-physics-land, where the conveyor is capable of infinite velocity, the wheels do not slip under any circumstances, and the wheels are capable of insanely fast rotational speeds without flying apart, it is possible to stop the plane from taking off. You can even assume zero friction.

But that's a very special set of circumstances. In pretty much all other (more realistic) cases, the plane takes off.

Correct, a small amount of energy is used to rotate the mass of the wheels. but wouldn't that energy be generated by the super fast super powered tread mill, and not by the plane? As long as the friction is zero on the axles the wheels could move infinitely fast with the tread mill, and the plane wouldnt move at all even under 0 power.


edit: IE, the wheels have a frictioned connection to the conveyer belt with the rubber touching rubber, but the ball bearing connection to the plane is frictionless, thus the energy and angular momentum provided to the wheels comes completely from the surface the wheels are touching.

 

[jagec]

Originally posted by: dxkj
Correct, a small amount of energy is used to rotate the mass of the wheels. but wouldn't that energy be generated by the super fast super powered tread mill, and not by the plane? As long as the friction is zero on the axles the wheels could move infinitely fast with the tread mill, and the plane wouldnt move at all even under 0 power.


edit: IE, the wheels have a frictioned connection to the conveyer belt with the rubber touching rubber, but the ball bearing connection to the plane is frictionless, thus the energy and angular momentum provided to the wheels comes completely from the surface the wheels are touching.

If you draw the FBD, you find that the conveyor exerts a linear force on the base of the wheel, and the engines exert a linear force in the opposite direction through the axis of rotation. This creates a torque. If the force of the engines is not balanced perfectly by the force of the conveyor, you'll get linear as well as angular acceleration. This would be the case in any real-world scenario, since it's nearly impossible to exert the same force on the wheels with the conveyor, as you can on the plane with the engines. F=ma; when you're working against the large mass of the plane, that's fine. When you're working against the small rotational inertia of the wheels, the acceleration term becomes exceedingly large.

But in magical physics land, where we can accelerate the conveyor as fast as we bloody well please, we're able to equalize the forces. Thus, there is no net lateral force, just a huge torque that's spinning the wheels faster and faster, until the plane runs out of fuel.

Even with frictionless bearings, the wheels are able to exert a lateral force on the plane through the angular momentum term. That's the secret. It's easier to think of a massively weighty wheel on a treadmill...if we exert no additional forces on it, the wheel will begin to slowly turn and slowly move backwards at the same time when we turn the treadmill on. If we add a force through the axis of rotation, we're able to keep the wheel in one place, but it builds angular momentum that much faster.

 

[chcarnage]

I remember the Ethanol + Water experiment from school. The teacher added 250 ml of one to 250 ml to the other and the resulting volume was something about 480 ml. It's not a chemical reaction but the smaller water molecules partly fill the space between the larger ethanol molecules.

 

[Kelnoen]

1 / 3 = a third
1 / 3 = 0.333...
a third times 3 = 1
0.333... times 3 = 0.999...
therefore
0.999... = 1

 

[Sc4freak]

1 - 0.999999.... = 0.00000 ...... 01
let 0.00000 ...... 01 = x
0.999999.... + x = 1
x = 0 / 10 + 0 / 100 + 0 / 1000 .... + 0 / (8-1) + 1 / 8 = 1 / 8
x = 1 / 8 = 0

0.999999.... + x = 1
0.999999.... + 0 = 1
0.999999.... = 1

I'm not sure if this proof I offered is valid, but 0.9 recurring is definately equal to 1.

Also, I have wikipedia on my side.

 

[GoodRevrnd]

Originally posted by: Kelnoen
1 / 3 = a third
1 / 3 = 0.333...
a third times 3 = 1
0.333... times 3 = 0.999...
therefore
0.999... = 1

That's bunk. 0.33333... is an approximation for 1/3. Therefore .999 doesn't = 1. At least by your argument.

 

[cirthix]

Originally posted by: GoodRevrnd

Originally posted by: Kelnoen
1 / 3 = a third
1 / 3 = 0.333...
a third times 3 = 1
0.333... times 3 = 0.999...
therefore
0.999... = 1

That's bunk. 0.33333... is an approximation for 1/3. Therefore .999 doesn't = 1. At least by your argument.

it's an approximation with infinite accuracy, if the approximation is taken to an infinitely large amount of decimal points.

 

[911paramedic]

I'm going back to bed, too much going on in this thread. :-/

 

[So]

Okay, this thread has officially been answered and derailed. And yes, the plane takes off, if you make assumptions that remove the completely illogical parts of the question.

 

[thesurge]

Originally posted by: GoodRevrnd

Originally posted by: Kelnoen
1 / 3 = a third
1 / 3 = 0.333...
a third times 3 = 1
0.333... times 3 = 0.999...
therefore
0.999... = 1

That's bunk. 0.33333... is an approximation for 1/3. Therefore .999 doesn't = 1. At least by your argument.

Infinite Geometric Series ftw.

.9999....=sigma(9/(10^k),k,1,oo)=\frac{(9/10)}{1-1/10}=1

 

[thesurge]

Informal proof of formula for infinite geometric series:


for -1<r<1
lim(n-->oo)[a(1-r^(n+1)]/[(1-r)]=a_1/(1-r)

 

[ultimatebob]

I'll believe that the plane takes off when they show it on Mythbusters. Until then, I'm calling shens on that. My hunch is that either the landing gear or the treadmill would fall apart from the stress before it got airborne. We don't live in "ideal Physics world".

 

[jman19]

Originally posted by: GoodRevrnd

Originally posted by: Kelnoen
1 / 3 = a third
1 / 3 = 0.333...
a third times 3 = 1
0.333... times 3 = 0.999...
therefore
0.999... = 1

That's bunk. 0.33333... is an approximation for 1/3. Therefore .999 doesn't = 1. At least by your argument.

No, it's not an approximation. 0.333... equals 1/3

 

[So]

Originally posted by: ultimatebob
I'll believe that the plane takes off when they show it on Mythbusters. Until then, I'm calling shens on that. My hunch is that either the landing gear or the treadmill would fall apart from the stress before it got airborne. We don't live in "ideal Physics world".

If you're accounting for friction in the wheels then you have to account for it in the treadmill, which would necessarily break long before the wheels did. Plus, the stupid question said the treadmill always traveled as fast as the wheels in the opposite direction, but the wheels will start moving when the plane does, so as soon as a delta v (acceleration) is introduced to the system, you have an illogical paradox where you tear a hole in spacetime and we all die. If you make the assumption that the user meant 'the treadmill is spinning as fast as the plane is traveling' then you never approach breakdown speeds of either and the plane takes off normally.

oy.

 

[mercanucaribe]

Originally posted by: DrPizza
Ummm, I'm going to have to disagree with the "mass is conserved" people as well. If a chemical reaction takes place, mass is lost to energy.

Mass isn't lost to energy. The product of a reaction has the same mass as the reactants.

Anyway, this has nothing to do with reactions.. If you mix some liquids, the volume will be less than the sum of the two.

 

[gsellis]

1ml of water added to 1ml of water = 2 ml. But then drop the room temp to < 0C and presto > 2ml of 'water'. Now if you mash 7.5 lbs of U235 into 7.5lbs of U235, if you could recover all of the pieces (and you did not do the mashing ), you would be missing at least 1 gram IIRC. Assuming you mashed them together hard enough. It was a long time ago when I heard it.

 

[mb]

I love how people think they are so smart (that wiki article is pathetic) because they have a few others that support their beliefs. I mean if I got a bunch of people on some random internet forum to tell you that blue = red, and created a wiki entry about it, does that suddently make it true? I can prove it! :roll:


There are several flaws with the bogus .9999... = 1 claim. One is that it can never be worked the other way. Ie, you will never get 2-1 = .9999... Say you have two apples and one is taken away. You have 1 apple left. You won't have some apple that is smaller by some infinitesimal minute amount.

Even the 1/3 + 1/3 + 1/3 example is bogus.
.33333.. + .33333.. + .33333.. will equal .9999999... the numbers don't round themselves up to 1, we do that.
3/3 will equal 1. You will not divide a number by itself and get anything but that number (save for 0). 100/100 = 1. 5/5 = 1. 89798798798708978907894867451387190970 / 89798798798708978907894867451387190970 = 1.

Same goes for multiplication. .99999.. times 10 will always be some value less than 10. Yes, it will be so close that it is practically indistinguishable, but it is not 10.

And addition. 1 + 1 = 2. 1 + .99999... will be some number smaller than 2. If you think 1 + .9999... equals 2, that's because YOU are rounding it up.

And subtraction. 1 - 1 = 0. 1 - .99999.... will be some number greater than 0. Yes, there REALLY will be a 1 trailing an infinite amount of zeros. Such a number is so small that it is hard to conceptualize, but that doesn't mean we can just say it is 0 because it's so small.

Even the limit example on wiki is weak. The problem effectively rounds .999... to 1.

Just because .999... is interchangeable with 1 in some problems does not make them equal. Hell, instant oatmeal + water + microwave = cooked oatmeal. Instant oatmeal + milk + microwave = cooked oatmeal. Does that make water = milk? Two different things that can cause the same result in some situations does not make them equal to each other in all aspects.

Flame away math geeks.