# WHY? is the relationship between speed and time exponential?

1. Aug 8, 2012

### DeepThoughts

WHY!? is the relationship between speed and time exponential?

I am actually a finance major but I like to learn about physics on my spare time. One of the more fundamental concepts I am having a hard time understanding is why the relationship between speed and time is exponential. Can someone explain this is simple terms please?

2. Aug 8, 2012

### Cerlid

Re: WHY!? is the relationship between speed and time exponential?

Not really what do you mean, why is time and speed exponential as a mass object approaches c, is that what you mean, even then it is not clear at all?

You may need to ask a more well defined question?

3. Aug 8, 2012

### ZapperZ

Staff Emeritus
Re: WHY!? is the relationship between speed and time exponential?

We are curious as well, because obviously, none of us know what you are talking about.

I can point out one relationship, v = s/t, where is the displacement. Where is the "exponential" here?

Zz.

4. Aug 8, 2012

### DeepThoughts

Re: WHY!? is the relationship between speed and time exponential?

Please excuse my vagueness. What im asking is, for example in a particle accelerator, when particles are sped up close to the speed of light their time relative to us becomes slower. What im asking is why the time dilation is greater in only the much higher percentages of the speed of light like 99.999% of C. From 1% to 99% there is not much of this effect. Why is this relationship exponential.

5. Aug 8, 2012

### ZapperZ

Staff Emeritus
Re: WHY!? is the relationship between speed and time exponential?

Can you please write down this exact relationship that you are referring to? That will be a very good and CLEAR starting point.

Zz.

6. Aug 8, 2012

### Cerlid

Re: WHY!? is the relationship between speed and time exponential?

Am I getting you wrong or are you asking why c is the speed limit of non mass objects such as the photon and why approaching c is the speed limit of mass objects?

What you are in fact asking is why the equations make c the constant it is or the limit of speed in time. This is a consequence of relativity and the laws of time, they are taken to be true because nothing contradicts them, they are derived as a consequence there of, not as something that comes from a principle that can be derived further than the equations and the resultant experiments we have at least tried themselves.

That is then not a question that is answerable, it just seems that as mass objects approach c, they experience a time dilation/length contraction that makes the concerns of speed relative to another co moving body always less than c in mass systems. It's a fundamental law of time, we don't know why the limit is c, we only know that asymptotically no mass object will ever reach it because the time and space issues forbid it.

It's not an exponential though, although I took you to mean you thought it was. Energy concerns blow up at speeds approaching c, but they are not e^x or an exponential concern in those equations.

EDIT:

Oh and what Zapper z said, let's start with a clear question, before we proceed to a clear answer.

Last edited: Aug 8, 2012
7. Aug 8, 2012

### Gravitational

Re: WHY!? is the relationship between speed and time exponential?

Do you know the lorentz factor and the time dilation formula? From the time dilation formula it is clear that the v^2/c^2 term is negligible for smalll speeds, as it will roughly equal zero. We then see that as the velocity approaches the speed of light, these changes become more apparent, and there is a much greater difference between the proper time interval and the observer's time. Hope i could answer you question. Maybe you can take a look at this graph also: http://www.thebigview.com/spacetime/timedilation.html

8. Aug 8, 2012

### DeepThoughts

Re: WHY!? is the relationship between speed and time exponential?

Actually Cerlid and Gravitational have provided the answer im looking for. In Gravitationals link and referral to the Lorentz factor in the time dilation formula, there is the exact equation and graph im trying to refer to. My question specifically I suppose is why is the relationship not linear. And the answer im gathering is "Because thats the way the math works out." and as Cerlid points out the reason for this is not answerable. Thanks for the patience with my intelectual shortcomings!

9. Aug 9, 2012

### Muphrid

Re: WHY!? is the relationship between speed and time exponential?

Every object has a path or trajectory through spacetime, and there are certain paths that do not change no matter how you move relative to the object on such a path--a path that light can take. Why the world works this way is big, big question, but if you can accept it on its face, then the formulas for how objects accelerate and behave follow.

10. Aug 9, 2012

### jbriggs444

Re: WHY!? is the relationship between speed and time exponential?

An "exponential" growth pattern is one in which something doubles in a fixed amount of time. And then doubles again in the same amount of time. And so on. This would include things like the growth of bacteria in a medium, the growth of money in your savings account or the splitting of atoms in an atomic bomb.

The term "exponential" is also used to describe decay patterns where something shrinks to half in a fixed amount of time and then half of that in the same amount of time. This would include things like radioactive decay or the concentration of a substance after multiple dilutions.

It is popular to use the term "exponential" to refer to things that grow "very fast". That usage is incorrect. (It tends to indicate either sloppy thinking or a desire to persuade without regard for the facts).

Some things grow without bound as some limiting value is approached. This would include things like the pressure in a tank as the volume shrinks to zero. That kind of growth is called "asymptotic".

Time dilation is grows "asymptotically" as relative velocity approaches c.

11. Aug 9, 2012

### ZapperZ

Staff Emeritus
Re: WHY!? is the relationship between speed and time exponential?

You should also learn something from this, that you need to be careful with how you describe things. Everything in physics has an underlying mathematical formulation. You simply cannot use terms such as "exponential" without understanding what it means! This is, after all, a physics forum, and when you use something like that, most of us know what it is. But you obviously didn't, but you are using it based on some mistaken idea. It creates a horrible confusion, not to mention, a lot of vague misunderstanding.

Zz.

12. Aug 9, 2012

### Staff: Mentor

Re: WHY!? is the relationship between speed and time exponential?

I actually think it is answerable. The math is founded on two postulates, first, the laws of physics are the same in all inertial reference frames, second, the speed of light is the same in all inertial reference frames. With mathematically precise expressions of those two postulates it is possible to derive the Lorentz transform, including the non linear relationship you noticed.

13. Aug 9, 2012

### Khashishi

Re: WHY!? is the relationship between speed and time exponential?

The relation cannot be linear because you can't reach infinity by increasing at a linear rate over a finite distance. Simply, if you divide infinity into equal steps from speed 0 to c, then that means the dilation is infinity/2 at c/2 speed??? That makes no sense, so the increase must be more than linear, and more than exponential even (which never reaches infinity over a finite domain). The actual relationship is something like
$\gamma=\frac{1}{\sqrt{1-\beta^2}}$
which goes to infinity as beta goes to 1.

14. Aug 10, 2012

### Cerlid

Re: WHY!? is the relationship between speed and time exponential?

If I may though the maths is far more important than the explanations I gave (gravitational hinted at a better explanation), you'll find out why things are what they are, and more importantly still is how the model applies to experiment. I just didn't know what level you are at so I didn't want to bombard you with equations.

If you really want to understand why c is the speed limit of the universe, then you are on the right track but the equations and the science will spell it out far better in terms of an overall understanding. You have some maths knowledge, if you're really interested, and have the time, there are worse things to do with maths. Economics for example, gah that's all unpredictable, physics is kinda tight, at least in special relativity.

Precisely get the model right and everything is a consequence of it. The beauty of it is everything is a consequence of taking something as being constant, it works because of that relationship as well in non inertial frames as it does in inertial ones, with c being that thing where everything else falls nicely into place from.

Last edited: Aug 10, 2012
15. Aug 11, 2012

### Neandethal00

Re: WHY!? is the relationship between speed and time exponential?

x = vt
v = x/t
If we keep x fixed and increase t, graph of velocity falls like an exponentially decreasing function.

See the JPG graph. Is this something you are referring to?
[sorry, can't insert a jpg picture]