1

New Theories / Re: How Many Numbers Exist?

« on: 15/10/2021 15:30:05 »

Hi.

I've Realized my capacity to understand & learn is Not infinite.

   Neither is mine.  I was especially "thick" as a child and couldn't read, write or do arithmetic until I was about 10 years old.  Fortunately the UK doesn't hold people back a year when they don't pass exams, so I did get out of primary school.

No, to divide through the Planck length infinitely many times would be equivalent to a singularity. That is itself equivelent to a breakdown in physics.

    This is about the limitations of our ability to probe below the planck length not the actual nature of space.

...The Planck length is expected to be the shortest measurable distance, since any attempt to investigate the possible existence of shorter distances, by performing higher-energy collisions, would inevitably result in black hole production....         [from Wikipedia]
    It doesn't follow that space isn't continuous, just that our current experiments to probe below that length are expected to create a black hole and change the nature of space in the region.

Best Wishes.

2

General Science / Re: How can photons have momentum but no mass?

« on: 15/10/2021 03:40:04 »

Hi.

The only thing (I think) I understood was that the question was inherently flawed

   I need to explain what I meant.

   There's no problem with your question.   There's no "right way" to ask a question, it's your question.  You should ask whatever it is you want to ask.

   When I said there's a flaw in the question, that's not your fault.  It's the way physicist use the term "rest mass" and all the things you would have seen elsewhere and in school.
    If you just pull the term "rest mass" apart as if it's a phrase in the English Language then it would mean the mass that a photon has when it's at rest.   However, this is not something sensible.   It would have been better if Physicist's had always called this thing  "the invariant mass" because then everyone would know it was something special with a precise definition.    However, that is not my decision to make.  Historically, "the invariant mass" has been called the "rest mass".   All we can do now is to make it clear that this isn't the mass that a photon would have when it's at rest.

   I've decided to cut down my original post, it was far too long and you probably aren't that interested in the details anyway.
   Special relativity demands that light will always travel at the speed of light, c  (in any inertial frame - but I'm not going to discuss inertial frames here - so I'll just say  "in all sensible situations").  So it is not possible for a photon to be stationary in any sensible situation.  In particular, we have no hope of measuring the mass of a photon when it is at rest because it just can never be at rest.
     Therefore the "rest mass" of a photon is just an abstract quantity and we should really call it the "the invariant mass" so that everyone knows it's an abstract, generalised or ideal thing and not something we can actually measure directly.

     The next important thing is that you have probably read or been taught that    momentum = mass x velocity.    This is probably why you are concerned about the photon having 0 mass   but still having a non-zero momentum.
     There are at least two ways we can address this issue.   The first is to say that many physicists were also troubled about this.  It's a very good question to ask and something that does seem quite puzzling.
      Physicist's were sufficiently determined to maintain this simple concept of momentum that they developed a quantity called "relativistic mass".   They accepted that the invariant mass of a photon wasn't anything you could ever really measure, it certainly wasn't going to be measured as the mass of the particle when it was at rest in some inertial frame.  So they determined that the invariant mass wasn't something that should be used in that formula     momentum = mv.     The "relativistic mass" is something you can almost measure, you can certainly infer it's value from some simple observations, and this is what you needed in that formula.    So we could answer your question by saying that although a photon has 0 invariant mass   (or  0 "rest mass"),  it has a non-zero relativistic mass  and that's what's needed to allow it to have momentum.

      More recently, the idea of "relativistic mass" has gone out of favour.  It's best if we just don't use it and only use the quantity called invariant mass.  In this situation we still have a problem  with the simple formula   momentum = mv   but we're not encouraged to modify the mass term, m in order to fix that equation.   We have to accept that the momentum is NOT given by the expression  mv.     Instead the momentum of a particle is obtained from a more complicated structure called the four-momentum.    You probably don't want a full explanation of what a four-momentum is  (there are references on Wikipedia and/or textbooks  and you could always ask if you really did want to know about this more complicated structure).
     Anyway, we could now answer your question by the second method:   We could state that momentum isn't exactly what you thought it was or might have been taught at school.  It never really was given by the simple expression    momentum = mv.   That expression is just an approximation that holds well for slow particles.   Momentum was really obtained from a more complicated structure, the four-momentum, and there was never any requirement for the invariant mass to be non-zero in order for a particle to have a non-zero momentum.
     
   Let's put this directly into action on the original question:

So... Photons lack mass (you know, the thing that gives an object its weight and the ability to impart a force onto another object), yet they carry momentum capable of... moving solid objects? What? How on Earth does this work exactly? How can something with no mass, carry momentum and more importantly, where does the momentum even come from?

  1.   Photons have 0 invariant mass but this is just an abstract thing.  We could never directly measure the mass of the photon when it's at rest.  So the mass of a photon is actually undetermined, all we know is it's "invariant mass" and, if we wish to calculate it and use it, "the relativistic mass".
  2.   Mass doesn't give a particle weight.   This requires gravity and gravity acts on any source of something we often call "stress-energy" or "energy-momentum"  -  mass is just one common example of this "stress-energy".  To phrase it another way, mass is NOT required.  Anything with momentum and energy will be a source of gravity and anything with momentum and energy will be influenced by gravity.
  3.   The ability to impart a force onto another object is granted by many things.  You were presumably talking about direct physical collision and the transfer of momentum.  This only requires that the incident object has some momentum and in the formal definition of momentum  (from four-momentum) this does not require the incident particle to have a positive invariant mass.  Non-zero mass is only required in the simple model of momentum that might be taught in school and/or applies adequately for slow particles.
  4.    "How on earth does this work exactly?"  It rests upon a decision to give up the school definition of momentum and recognise that it has limits.  You can't save the school definition of momentum = mv  when considering fast particles.  You can either "tweak it" by using a quantity called "relativistic mass" or else directly utilise a better definition of momentum.

Best Wishes.

3

New Theories / Re: How Many Numbers Exist?

« on: 14/10/2021 14:08:57 »

Hi again.

I've got to ask:   Do you know this Mathlogger person?  Why are you promoting these videos?   They're good - but what is this for?  It takes time to watch these and you don't seem to want to discuss anything about them.

Best Wishes.

4

New Theories / Re: How Many Numbers Exist?

« on: 14/10/2021 13:05:24 »

Hi.

Indeterminate: the hidden power of 0 divided by 0 .........  (and video provided)...

I can review the video, if that helps:
    It was a 12 minute video.  You can safely watch it in 6 minutes if you choose double-speed.  Calculus was discussed and L'Hopitals rule.   Something controversial was said about 5 seconds before the end, so you may like to slow it down for that.  If you're new to calculus, this video might provide some useful graphics and explanation.  If you're interested in L'Hopitals rule the video doesn't provide enough detail to help much.

Best Wishes.

5

New Theories / Re: How Many Numbers Exist?

« on: 11/10/2021 23:15:38 »

Hi @Zer0 ,

SideNote - i am a self proclaimed & self diagnosed entity who has a self proclaimed Myself unfit for the Subject.(OP)

  Self-diagnosed?  I don't know where you live but if it becomes useful or relevant you could get a formal test and diagnosis.  It could be that something simple can be done.   Maybe all you need are some coloured glasses and some classical music playing in the background,  or less of something you're normally exposed to (like caffeine or some other substance).

Different Individuals with different backgrounds & identities...moving forward towards a common goal.

   Hopefully, that's what should be happening in most forum threads.

   Anyway, I wish you well, Zer0.   Bye for now.

6

General Science / Re: Questions that Antimatter to Me

« on: 08/10/2021 16:17:02 »

Hi @Aeris
   No worries.  I was just checking you're happy enough with the answers.
Good Luck with your college work and don't undersestimate your own ability to understand Science.  You're asking good questions and one day soon you'll be teaching it to someone else, writing for a science magazine or whatever it is you want to do.
Best Wishes.

7

New Theories / Re: How Many Numbers Exist?

« on: 06/10/2021 02:27:28 »

Hi.

Many arithmetic procedures break down when infinity is involved.  Otherwise we would prove that 1=2.

  Agreed.

I think it's easier and simpler to analyze it in binary form, since we only need to consider two possible values in each digit.

   O.K.

If every possible combination of 0 and 1 is already covered in a set, then asserting that there are other sets containing other combinations not found in the first set is absurd.

  Agreed.  If it is true that the original set did cover all possible combinations of 0 and 1, then there can't be any other set containing other combinations not found in the first set.

    However, the diagonalisation argument is used in a proof by contradiction.   We aren't arbitrarily "asserting" that there other sets containing other combinations, we prove that there would be   AND  we don't really believe that the original set did cover all combinations of 0 and 1.  We are just temporarily assuming that the original set covered all possibilities just to get set-up for a proof by contradiction.  After this we will KNOW that the initial assumption we made had to be false.  So the original set didn't really cover all possible combinations of 0 and 1.

Example
   One of the easiest examples of a proof by contradiction is the proof that √2  is irrational,  this is often seen at school.
Here's one website that presents the proof -  https://www.mathsisfun.com/numbers/euclid-square-root-2-irrational.html

   The typical use of the diagonalisation method is in a proof by contradiction.  For example, this is how it is used to show that there isn't a bijection between ℵ and  the interval [0,1] :      Temporarily assume we have a bijection 
F:  ℵ → [0,1].  Then construct a test number, T,  so that ........etc.......  Ultimately we have T ∈ [0,1] but T ∉ Range(F).  So F was NOT a bijection.     
   This is a contradiction, so our initial (temporary) assumption had to be false.     Now we  KNOW  that there can't be a bijection  from ℵ to the interval [0,1].

- - - - - - - - -
    Sometimes it's best to say things in two different ways.  Here's another approach:

If every possible combination of 0 and 1 is already covered in a set, then asserting that there are other sets containing other combinations not found in the first set is absurd.

    Agreed.   The diagonalisation method can't be used to show the first set didn't cover all combinations of 0 and 1.  If you define the set to be all possible combinations, then that is what it is.    The diagonalisation method would only force the conclusion that this set cannot be "enumerated".   It cannot be put into a 1-to-1 correspondence with the Natural numbers.

Best Wishes.

(Late editing - to try and keep mathematical expressions together on one line and not split over two lines)

8

New Theories / Re: How Many Numbers Exist?

« on: 03/10/2021 21:40:57 »

Hi again.

perhaps we can just send it to ArXiv, or it's alternative like Vixra. Considering the widespread interest on this subject, I think once published, it will reach wider audience in no time.

   Arxiv will not take articles that are submitted by unregistered authors UNLESS these articles are endorsed by others (who are already registered authors).  It's not as simple as sending a copy to Arxiv by mail and expecting them to make it available to the world.  The usual way to obtain endorsement is to be affiliated with a recognised University or research establishment.  To the best of knowledge, the Naked Scientists forum is not one of those recognised establishments.

    Vixra is an alternative but take some time to read what they say about themselves:
ViXra.org is an e-print archive set up as an alternative to the popular arXiv.org service..... It has been founded by scientists who find they are unable to submit their articles to arXiv.org because of Cornell University's policy of endorsements and moderation designed to filter out e-prints that they consider inappropriate....
   ViXra is an open repository for new scientific articles. It does not endorse e-prints accepted on its website, neither does it review them against criteria such as correctness or author's credentials.   -  taken from Vixra website,  foot note on their main page.   (Active link omitted since I can't even recommend that you visit.  No one is entirely sure who pays for or maintains this website).

   It takes only a moment to realise that there are very few quality checks on anything published on Vixra.   There are some people who have gone so far as to say the following:....
Warranted or not, it has a reputation of being an alternative to arXiv for cranks and to host a lot of junk science, fake proofs or even outright nonsense. It is not recommended to submit something to viXra if you wish to be taken seriously, because the reputation of viXra would probably taint your paper's reputation by association.    
- Taken from Academia stackexchange website.  https://academia.stackexchange.com/questions/105518/what-are-the-differences-between-vixra-org-and-arxiv-org

    Even if you did get endorsement from someone and got a paper to Arxiv,   this still doesn't count as any formal publication.  The articles and papers on Arxiv are not peer-reviewed in the same way as more conventional papers published in journals.   More importantly there are thousands of articles on Arxiv and even they (Arxiv) explain that there is no reason to expect anyone else to find or see your article.   You have to actively go out and make contact with others working in the field and invite them to look at the article.  Only after your article has been cited a few times does it stand any chance at all of being found by other academics or the sort of people who you are trying so hard to help.
   Even after this, the only way to really get it into the mainstream of education and further research is to get it properly published and subjected to a peer-review process in some proper journal.   Most professionals do not have the time to study new articles on Arxiv, they barely have enough time to review articles that do get published in journals.
    I can only apologise for writing so much, it's just that publishing a research paper isn't a simple matter.

Considering the widespread interest on this subject, I think once published, it will reach wider audience in no time.

    There is no history that I am aware of where something like a New Theory article presented in a forum has become a shortcut to getting a proper research paper published.  I doubt it will get published to begin with. 
    Forgetting about outside interest, the interest here in this forum doesn't seem to be as "widespread" as you might have hoped.  I'm really sorry but maybe Maths isn't to most people's liking, or maybe there just isn't more than half-a-dozen people here in the forum on a typical day. 
- - - - - -

Anyway, I've already written too much and it's getting boring, sorry.   I'm just asking you to take a moment to evaluate the situation.
You said....

Perhaps we can collaborate to write a proper research paper on this.

   I'm guessing you were talking to me.  Let's say yes, yes I would.  How is it going to be possible?

    I'll answer some of the other subject-related questions or ideas you had in another post,  I've already gone on too long here.  Using the forum to discuss things seems useful and if you have a genuine interest in this subject, great, let's discuss it.

Best Wishes.

9

General Science / Re: Do the theories of quantum and relativity have to be unified?

« on: 02/10/2021 23:32:34 »

Hi.

We aren't going to lose anything we already have if we don't unify these theories.

     There is an obvious problem with physics in that relativity and quantum mechanics don't seem to match up and work well together.  They're both good theories which are pillars of modern physics.  So trying to get them to work well together is a sensible and natural thing to want to do.   So it's not a waste of time and effort, it is  "a big elephant" and it is right in the way of general developments in physics, rather than being some problem that Physicists have have gone looking for.  If we are only interested in developing our field of knowledge then this problem is a sensible one to analyse.
     Based only on history, advancements in theory do usually lead to all sorts of practical discoveries and new technologies.  However, on the short term, it's hard to see what practical benefit would come from a unified theory of quantum mechanics and relativity.  We would be able to analyse and hypothesize more on the early universe, understand black holes better (and other things that don't usually matter in your day-to-day life).   It's unlikely that this will quickly be turned into some new technology that gives us a better washing machine.   There's no piece of technology that I know of that is desperately waiting for a unified theory of quantum gravity - but maybe others on the forum will know of something.

Best Wishes.

10

New Theories / Re: How Many Numbers Exist?

« on: 02/10/2021 21:52:50 »

Hi again.

It's mind boggling that many brilliant mathematicians over a century failed to see this, while a few random forum members can solve it in just a few posts. Perhaps we can collaborate to write a proper research paper on this.

     Since no other random forum members seem to be interested, I suppose I've got to ask.   What did you have in mind?  ...and is there any food or light refreshements provided because it might influence my decision.

Best Wishes.

11

General Science / Re: Super conducting magnets and bigger objects.

« on: 01/10/2021 23:16:28 »

Hi.

You (@evan_au ) seem to be asking for a check on the Maths...

The Charge/mass ratio is 40% of a proton.

Seems OK.

So in the same electric field, the lead nucleus will have 82 times more kinetic energy of the proton, but will experience only 40% of the acceleration and will reach  36% of the speed (if I've done the sums correctly).

  I don't know.   Is the lead nuclues in the same electric field for the same TIME as the proton,   or  so that it covers the same DISTANCE  as the proton  covered?
    I reckon you're using    acceleration =  (Electric field Strength) . ( Charge )  /  (Mass).
   So, we have    accel. of Lead =     40% of  accel.. of proton         in the same   E  field.

After that.... we need to know something else,   is it accelerating for the same time OR over the same distance?   If it's allowed to stay in the ring for as long as you like and do as many loops as required, then anything goes.   The bending magnets are the limiting factor as you said later in your post.
- - - - - - -
None of this affects the spirit of what you've said and the rest of the post is good.

Best Wishes.

12

New Theories / Re: How Many Numbers Exist?

« on: 01/10/2021 13:42:49 »

Hi again.

   Like the many short articles for the general public, they miss out several fine details.
Here's one example:
   Suppose you had an enumeration of the real numbers that looked like this:

1  →  0.099999.......     (recurring digit 9)
2  →  0.19378652...
3  →  0.4793656.....
4  →  0.39592478...
............
The first number has recurring  digit 9,   it is 0.0999999...
The next numbers in the enumeration need to have the digit 9 in the n^th  decimal place  BUT all other digits are arbitrary. 
   Now build your test number as directed in the article (add 1 to the digit and note that 9 gets taken to 0), I've made the appropriate diagonal digits glow,
  Our test number = 0.1000000....  =  0.1
    However,  you might have seen the mathematical result that shows 

So that our test number, 0.1  is precisely equivalent to  0.0999999999.....  which was the first number in the enumeration.  So in this enumeration, the construction has failed to produce a number that wasn't on the list.  We would be unable to conclude that this enumeration wasn't a total enumeration of the real numbers.

   The requirement that there is a 9 in the (n)^th decimal place isn't as restrictive as you might think.  We could still argue that all real numbers between 0 and 1 might lie in that enumeration.   Consider, for example, the number   0.1239321.   It might seem that it has only has one 9 digit in the 4th decimal place and so it would have to be the number 4th on our list but this isn't true.   It's the same number as  0.123932099999999999999999....     (replace the last digit 1 with infinitely recurring 9) - so that it could be anywhere on the list from the 8th entry on the list through to the ∞ entry on the list.

    There's several places where the article has rushed or glossed over some complications that arise but it does what it set out to do well enough.  It's an interesting article that appeals to the general public, keeps them on the edge of their seat and more importantly makes the reader feel that they have understood something.  We all tend to learn more and feel more motivated by successfully understanding something.  It should help to get people interested in Mathematics, so thanks for presenting it here.

Best Wishes.

{Late editing:  Fixed some mistakes over n^th  and (n+1) ^th  decimal place.}

13

Just Chat! / Re: Not so good YT videos.

« on: 30/09/2021 12:15:32 »

Hi.

What do U, guys, think of all those physics videos, on YT?

   There's lots of them.  Some of them are quite informative others are just entertainment or  "info-tainment" as they are sometimes called.   Many of them are awfull.   Money and payment for getting views has become a huge part of YouTube.
   I don't think I can discuss the good ones in this thread.  The thread could quickly be taken over by people just advertising their own videos and/or those of friends.  The forum has a policy on advertising (try not to do it).  All links should be relevant to the discussion at hand, if they were more relevant to boosting your own (or your friends) profile or bank balance then they they really don't need to go in the posts.
   This thread was originally set up to discuss the problems in YT videos, in the hope that no one would want to put their own video in here anyway.

   There are many times when I would like to see, listen and even talk to someone, rather than just read something in a book.  YT videos are one of the few places where you can get something like this if you are unable to physically attend a University or similar place.  Forums can also be useful for this.

Best Wishes.

14

New Theories / Re: How Many Numbers Exist?

« on: 29/09/2021 23:06:00 »

Hi Zer0,

Isn't " Infinity " Not a value, but just a never ending process.

  Yes, sometimes.   Overall this would be a better way of thinking about it.  Well done.

Best Wishes.

15

Just Chat! / Re: Why don't people Say Thanks when they read a post?

« on: 28/09/2021 00:23:12 »

Hi.
     Possibly because there could be an entire page of posts saying "Thanks, I've read this" before any new post is reached which has some substantial content.
     Anyway, thanks for your post, I have read it and it's useful.   If it's important to you to know how many times a post has been viewed, I think it's on the statistics listed on the main page.
     It's potentially dangerous to worry about it.   People will "game" the system if it ever became important to do so, which can be done by various methods that I won't even mention here since I have no wish to encourage it.  Fundamentally, there's no reward, or point, in trying to be the most viewed thread.   
   If you meant using the "say thanks" action that drops down from the top right.  Different people use it in different ways.  Personally, I don't like using it much, it seems to be shortcut to spending time and writing something.  It's only worth using if you had nothing of any substance to add to the discussion but wish to convey some recognition or approval of the post for some reason.   There's no reward for having or giving a a "say thanks" either as far I know.  It's like getting a birthday card from someone who has just written their name at the bottom of it and no message, although there isn't even a nice picture to look at.

Best Wishes.

16

General Science / Re: What is Energy? Like, seriously, what is it?

« on: 26/09/2021 00:07:53 »

Hi.

When you say ideas, do you mean hypothesizes based on our current understanding of physics? Because if you do (and I cannot stress this enough), I will happily accept the most scientifically-sound/agreed-upon hypothesis as an answer.

   No,  I meant there is some history, various ideas that are now discredited, some sideline discussions that will surely pop up along the way,  more general discussion around the topic   etc.
   If you are seeking short and specific answers, perhaps you should flag that in bold  or  NEON - I am seeking the shortest and most current answer please.    (Although, t.b.h. there are many others who do just give short answers all the time,  I honestly don't know how they do it and how they can sleep at night).

Since I have so many more questions similar to this one, can I post all of them at once on a single thread? Like, is that allowed? Asking multiple questions in a single thread?

   I'm not a moderator but the following seems reasonable:
1.  It's not a good idea to post multiple questions on the same thread, especially if they are easily separated. 
  No one can read posts of hundreds of pages and the original title won't attract people who could answer the latest questions anyway.  Most people like self-contained bite-sized chunks of discussion.

2.  Provided you don't spam the site but are just keen to get answers to questions and/or discuss things sensibly and/or  answer other peoples questions  ... then this site is quite tolerant.  Many threads do have multiple questions and side-line discussions that pop into them.
    My personal advice is to be careful about "hijacking" someone else's thread completely - check that they are OK with whatever the thread has turned into.  Where possible, if your questions are easily separated then it is sensible to separate them.

Best Wishes.

17

General Science / Re: What is Energy? Like, seriously, what is it?

« on: 25/09/2021 14:57:55 »

Hi.

I'm starting to think that maybe I'm asking too many "We honestly don't know." questions on here...

   The thing is, you ask them well.  It seems (to me anyway) that you didn't want a short definitive answer and some of us prefer to discuss rather than lecture.

If you want the shortest possible answer then it's something like this:
    Energy is just a number.  Something we can calculate as a relation between some variables in a system.

If we go back to your original post and take a hard line approach to it then this is wrong:

Photons, which are LITERALLY made of energy....

    Photons are NOT made of energy.  They are NOT a pure or naked form of energy, there just isn't one.   We have never isolated a quantity of stuff we can call "energy",  it's always something in a particle, carried by a particle or possessed by a whole system.
    Photons are not energy they are something else,   they are oscillations in the electric and magnetic field if you want to see it that way and they transfer energy because those electric and magnetic fields interact with other things.  They have energy associated with them but they are NOT energy.
    All attempts to consider energy as "stuff" or some sort of substance have failed.  It is just a number, something we can calculate.

   Thanks for asking good questions and phrasing them well, sorry if some of us (that'll probably be me) say too much, rather than firing out a quick answer. 
   
Best Wishes.

18

Physics, Astronomy & Cosmology / Re: If the earth stopped orbiting the sun?

« on: 25/09/2021 14:31:50 »

Hi again.

This is a hypothetical question and may have an interesting answer.

   It may have an interesting answer?  You're not planning to try it are you?  We can't encourage these kinds of experiments.

Best Wishes.

19

Physics, Astronomy & Cosmology / Re: If the earth stopped orbiting the sun?

« on: 25/09/2021 14:27:13 »

Hi again.
  Well, if Halc's escape velocities are right   (i.d.k.  he probably found them in Wiki),  then that would do the job for you.
Those escape velocities would have been determined using the potential energy and kinetic energy forumulae mentioned earlier.
   What did he say? 

575 km/sec in fact which is the escape velocity of sun minus escape velocity of solar system from Earth. This is about 2 million km/hr or 1,200,000 mph

    There's a risk he was using a complicated solar system with all the masses of the planets, asteroids and whatever but let's just assume he was considering the sun as the only mass of interest.   Never do a claculation yourself if someone else will do it for you.

Best Wishes.

20

Physics, Astronomy & Cosmology / Re: If the earth stopped orbiting the sun?

« on: 25/09/2021 13:54:51 »

LATE EDITING:   The elliptical orbit shown is incorrect, sorry.  I assumed a symmetry where there isn't one.   See post #28 for a correction.


Hi.

    Well, I'm tired but I made it about 91 days until the earth hit the sun (about one-quarter of a year).  This is a simplification assuming the sun is just a point mass.  Realistically, it has some radius and we want to know where you consider the earth to have actually hit the sun.  It has a diffuse outer layer, so where do you decide the sun is solid enough (or hot enough) to say the earth has been destroyed?  I'm also completely ignoring drag, the space around the sun should start becoming quite dense and offering quite a lot of drag.  These are all fairly minor considerations because the heat from the sun is going to be the biggest problem.
    The most interesting thing is how you could determine this.  There are two reasonable methods:

1.    Differential equations:    We have,
     from  Newton     acceleration = gravity
This d.e. is a bit tricky but can be solved.

2.   A faster or slicker method is to realise the Kepler's thrid law will apply.   Falling to the sun in a straight line is just an extremely degenerate elliptical orbit.    Just consider that earth is allowed to have a small transverse velocity instead of being forced to have 0 transverse velocity, so it will just miss the sun and go in orbit like this:


ellipse.jpg (18.64 kB . 897x176 - viewed 528 times)

Anyway,  the semi-major axis ≈  1  A.U. ≈ the same as the semi-major axis in our current orbit.   So the total Time period ≈ the same as our current orbit = 1 year.
   The entire motion is extremely symmetric so we can split it up into quarters (it's symmetric about the centre in the x-axis and the y-axis), so the earth starts from the furthest point away from the sun (far left on our diagram) and comes closest to the sun after one-quarter of a time period (about 91 days).   For the extremely degenerate orbit (the straight line into the sun rather than a narrow ellipse around it) the earth would have hit the sun then.
    (Bored Chemist has written about viewing the orbit of the earth from a side profile..... it's actually very much like the idea presented above).

3.   Other methods probably exist. 


    Anyway, other articles consider the journey toward the sun to be over in about 65 days but that seems to be because they reckon the tidal forces from suns gravity would rip the planet apart before we even reach the outer edges of the sun.
See:  What Would Happen if the Earth Stopped In Its Orbit?   https://www.wired.com/2014/12/empzeal-earthfall/
for a day-by-day diary of what could happen on earth as we approached the sun.

As for the velocity at the point of impact,  I reckon it would be about ... well I can't be bothered to work it out.  You can do this yourself  Just Thinking:
    The potential energy of the earth is       using  Newton's gravity which will be more than good enough for us.    Find G = universal grav. constant in Wikipedia.      M = mass of the sun  (also in Wiki).
r = distance from the sun to the earth, this is going to vary as the earth moves closer to the sun.
    Set r = average distance from sun to earth  initially =  150 milion Km.
    Then determine the pot. energy at somewhere we can call the edge of the sun  --->  radius of the sun is about 696 thousand Km.

   The general idea is that the difference in potential energy is converted to kinetic energy.   
use  kinetic energy of earth =      with  m = mass of earth   (find it in Wiki).   and v = velocity of earth.

I don't have a calculator and I usually get the things wrong anyway since I have fingers that don't do what they are told.   Alancalverd is an engineer, he might enjoy doing the calculation, ask him/her if you can't do this yourself.

Best Wishes.

Late editing: Inserted diagram, fixed an error in a formula.