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Topic: A Scaredy Cat. (Read 3378 times) |
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rloginunix
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A Scaredy Cat.
« on: Jan 26th, 2014, 11:42am » |
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A Scaredy Cat.
Someone quietly snuck up behind a cat, tied a bunch of empty soda cans to her tail and gave her a gentle nudge. As the cat moved, the cans made some noise. The cat got scared and moved a little faster. The faster she moved the more noise the cans made. The more noise the cans made the more scared the cat got. The more scared the cat got the faster she ran. The faster she ran the more noise the cans made... Bottom line - the cat accelerates.
When will the cat stop accelerating?
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towr
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Re: A Scaredy Cat.
« Reply #1 on: Jan 26th, 2014, 12:29pm » |
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The noise won't scare her anymore once she breaks the sound barrier.
But in practical terms, she'll pass out from exertion long before she reaches that speed.
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rloginunix
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Re: A Scaredy Cat.
« Reply #2 on: Jan 26th, 2014, 1:18pm » |
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towr is correct.
On the practical side I guess I should've mentioned that this is purely a thought experiment. Not sure though, may be that would've been a giveaway.
A similar follow up if anyone is willing.
You are moving at the speed of light while looking in the hand held mirror in front of you. What would you see in that mirror?
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rmsgrey
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Re: A Scaredy Cat.
« Reply #3 on: Jan 27th, 2014, 7:26am » |
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on Jan 26th, 2014, 1:18pm, rloginunix wrote:towr is correct.
On the practical side I guess I should've mentioned that this is purely a thought experiment. Not sure though, may be that would've been a giveaway.
A similar follow up if anyone is willing.
You are moving at the speed of light while looking in the hand held mirror in front of you. What would you see in that mirror? |
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If you take account of Special Relativity, then, so far as you're concerned, no time is passing while you're at light-speed, so you wouldn't see at all.
If you don't, then you need to be clearer about your modelling assumptions - if I turn on a torch while walking at 1 m/s, does the light come out at 3e8 +1 m/s relative to someone standing still, or at 3e8 -1 m/s relative to me? If light travels at a constant speed relative to its source, then you'd see a normal image of yourself in the mirror; if it's at a constant speed relative to a fixed aether, then you wouldn't be able to see the mirror, let alone any image in it because no light would be able to catch its near side in order to bounce off...
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rloginunix
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Re: A Scaredy Cat.
« Reply #4 on: Jan 27th, 2014, 10:56am » |
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Yes, need more context. Sorry about that.
I saw this question on a PBS Nova show about A. Einstein. The context was his thoughts prior to his 1905 Special Relativity paper. He worked as a Swiss Patent Office clerk then. Looking at the nearby clock tower he questioned "what will happen if one were to move away from the clock while still looking at it"? "What will happen if one were to hop on a ray of light"? Next, the narrator mentioned the hand held mirror question. They gave no answer but showed Einstein holding a mirror with blackness in it. So I think we have to consider the Special Relativity.
The current wisdom is that no material object can reach the speed of light. The mass will be infinitely large and the time will stop. But as a thought experiment I would imagine a point light source "at rest" and two primitive cases. The mirror moving towards and away from it.
(Point light source shines a narrow ray of light in one direction along which the mirror is moving).
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rloginunix
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Re: A Scaredy Cat.
« Reply #5 on: Jan 28th, 2014, 7:09pm » |
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Quick update.
Not sure if that's exactly the episode I was talking about but in the link below they discuss this at 54:00:
Einstein's mirror
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| « Last Edit: Jan 28th, 2014, 7:11pm by rloginunix » |
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JiNbOtAk
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Re: A Scaredy Cat.
« Reply #6 on: Feb 10th, 2014, 8:33pm » |
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on Jan 27th, 2014, 7:26am, rmsgrey wrote:| If you don't, then you need to be clearer about your modelling assumptions - if I turn on a torch while walking at 1 m/s, does the light come out at 3e8 +1 m/s relative to someone standing still, or at 3e8 -1 m/s relative to me? |
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I've always wondered about this, which is actually true? When we say light is moving at a constant speed, constant to what exactly?
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| « Last Edit: Feb 10th, 2014, 8:33pm by JiNbOtAk » |
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Grimbal
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Re: A Scaredy Cat.
« Reply #7 on: Feb 11th, 2014, 3:08am » |
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Constant is constant. If it is relative to something, it is not really constant, isn't it?
Well, ok, it is constant to the observer, regardless of the referential. If you move to near the speed of light, you will still observerve the light moving forward or backwards at the same speed, i.e. use the same time to cover a given distance.
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rmsgrey
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Re: A Scaredy Cat.
« Reply #8 on: Feb 11th, 2014, 6:54am » |
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on Feb 10th, 2014, 8:33pm, JiNbOtAk wrote:
I've always wondered about this, which is actually true? When we say light is moving at a constant speed, constant to what exactly? |
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That was one of the big questions in theoretical physics around 1900 - the leading theory was that, just as sound waves travel through air, light waves travel through "aether" - and the speed of light is relative to that aether. Experiments were conducted to detect the Earth's motion relative to the aether and they consistently said that the Earth wasn't moving relative to the aether. Discarding the possibility that the Earth is actually motionless while the rest of the universe revolves around it, other explanations were tried - like motion through the aether changing the lengths of objects to cancel the effects experimenters were looking for.
Einstein's moment of genius was to say "what if, rather than the speed of light being fixed relative to something, we take the experimental results at face value and say that the speed of light is fixed relative to everything - that, so long as you're not accelerating, you will always measure the speed of light as the same relative to yourself as any other non-accelerating observer?" and then followed through on the consequences of that - length contraction, time dilation, mass increase... Pretty much all the things the aether would have to cause in the models where the aether makes itself undetectable. So "aether" was quietly dropped from the models in favour of "relativistic space-time"
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rloginunix
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Re: A Scaredy Cat.
« Reply #9 on: Feb 11th, 2014, 7:07am » |
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That's right. Einstein captured the constancy or invariance of the speed of light in the Second Postulate of his Special Theory of Relativity - the speed of light in vacuum for all the inertial frames of reference is the same (regardless of the state of motion of the light emitting body or the observer).
In plain English it means that if a flash light is resting relative to you, moving away from or towards you no matter what measurements you make you will always come up with the same number known as c.
You may observe a change in frequency of light - redshifting if the source of light is moving away (larger wavelength, smaller frequency) or blueshifting if it's moving towards you (smaller wavelength, larger frequency) but its speed will always be the same.
The mathematical apparatus that supports this idea is known as Lorentz transformations and the formula for the resulting velocity of two moving items (in vector form) is:
r = (v1 + v2)/(1 + v1*v2/c^2)
So if you, for example, simply replace v1 and v2 with c, as a limiting case, you will still get c as a result.
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rloginunix
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Re: A Scaredy Cat.
« Reply #10 on: Feb 12th, 2014, 2:08am » |
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Back to the mirror problem.
Though technically and practically unachievable these days the following rewording is more scientifically correct:
A spaceship is flying in a straight line at constant speed of 0.9c relative to planet Earth. To eliminate gravity massive objects are absent. Inside the spaceship in complete darkness an observer (a person) is holding a mirror in such a way that the line connecting the observer's eye and the mirror is parallel to the ship's velocity vector.
A flashlight located near the observer's head is turned on.
Will the observer see his/hers reflection in the mirror?
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rloginunix
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Re: A Scaredy Cat.
« Reply #11 on: Feb 12th, 2014, 2:16am » |
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The answer is yes, he/she will. Reason being is the invariance of the speed of light discussed above.
Let's say the distance between the eye and the mirror is 30 observer's cm and the speed of light is 300 000 km/s. Then in 1 nanosecond of observer's time the light will reach the mirror. Another 1 nanosecond later the reflected light will reach the observer's eye.
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JiNbOtAk
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Re: A Scaredy Cat.
« Reply #12 on: Feb 12th, 2014, 4:24pm » |
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rmsgrey <like></like> (Ok, so we don't have a like button)
Seriously man, thanks!! I had to read your response a couple of times to get the idea, but it's a much better explanation than I've gotten elsewhere.
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Grimbal
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Re: A Scaredy Cat.
« Reply #13 on: Feb 13th, 2014, 8:31am » |
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In fact, the observer won't notice anything different at 0.999c compared to when he's still.
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rmsgrey
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Re: A Scaredy Cat.
« Reply #14 on: Feb 20th, 2014, 6:49am » |
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on Feb 13th, 2014, 8:31am, Grimbal wrote:In fact, the observer won't notice anything different at 0.999c compared to when he's still.
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Unless he looks at the rest of the universe, of course...
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rloginunix
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Re: A Scaredy Cat.
« Reply #15 on: Mar 4th, 2014, 6:14pm » |
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And to bring this to a somewhat logical conclusion let's see what a side, Earth-based, observer will witness.
For him/her the spaceship's 30 cm will Lorentz-contract to 30*sqrt(1 - v^2/c^2) = 30*sqrt(1 - 0.81) = 13.076cm. The distance will shrink by more than twice. The wrinkle with timing of things, however, is that while for the spaceship's person nothing but the light moves for the Earth-based observer the mirror is also moving - it's trying to run away form the first photon produced by the flash light but it's fighting a losing battle. It will take quite a while for a photon to catch up to the mirror (despite the distance contraction) and it should be a very short time for the reflected photon to reach the spaceship person's eye.
To calculate the times in this case we need to find the speed with which the photon is moving relative to the Earth-based observer. Of course we know that it is C but for demonstration purposes let's do it the formal way. If v1 is the speed of the spaceship's FoR relative to our (Earth-based) FoR and v2 (actually c) is the speed of a photon in the spaceship's FoR then the speed of the same photon relative to our FoR is:
(v1 + v2)(1 + v1*v2/c^2) = c*(v1 + c)/(c + v1) = c
As was expected. Before proceeding with calculations though, according to Einstein, we should make a fearless prediction that the total photon's round trip travel time for the Earth-based observer must be longer than 2 nanoseconds by exactly as much as 30cm is larger than 13.076cm. Let's see if this time slowdown is true.
From that point on it's 5-th grader math. For the photon catching up to the mirror:
v2------D---- v1-----x-----C
C is the point where the photon catches up to the mirror and D is the initial distance between the photon and the mirror. v2 = c, v1 = 0.9c, D = 13.076cm. The mirror will cover the distance x = v1*t1 while the photon will cover (D + x) = v2*t1, where t1 is the time it takes the photon to catch up to the mirror. Two unknowns, two equations. Solve for x, then for t1.
x = D*v1/(v2 - v1) and t1 = D/(v2 - v1) = 13.076/(0.1*3*10^10cm/s) = 4.358*10^-9s or 4.358 nanoseconds.
For the reflected photon v1 is the speed of the eye (0.9c), v2 is the speed of the photon (c) moving towards the eye:
v1------M----------x------------- v2
Here M is the point somewhere within 13.076 cm where the spaceship person's eye and the reflected photon meet:
x = v2*t2
D - x = v1*t2, where t2 is the time it took the eye and the photon to meet.
x = D*v2/(v1 + v2) and t2 = D/(v1 + v2) = 13.076cm/(1.9*3*10^10cm/s) = 0.229*10^-9s or 0.229 nanoseconds.
The total round trip travel time is t1 + t2 = 4.358 + 0.229 = 4.587 nanoseconds. 30cm/13.076cm = 2.294 and 2*2.294 = 4.588 nanoseconds which matches our calculated time of 4.587 nanoseconds closely. So the spaceship person's 2 nanoseconds become 4.587 nanoseconds for the Earth-based observer.
[edit]
Fixed a typo: 0.029 was meant to be 0.229 which is what it's now.
[/edit]
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| « Last Edit: Mar 5th, 2014, 12:33pm by rloginunix » |
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UgoLocal02
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Re: A Scaredy Cat.
« Reply #16 on: Jun 12th, 2014, 12:52am » |
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in simple terms the cat will stop when she is tired, and when she stops she stops accelarating.
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Annettagiles
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Re: A Scaredy Cat.
« Reply #17 on: Sep 29th, 2014, 4:32am » |
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until the sound stops the cat will not stop accelerating
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