Trying to derive the time dilation formula

[Mark]

Hey everyone, I am doing relativisitc physics right now, and in my notes the following formula was derived.

$$t' = t\sqrt{1- \frac{v^2}{c^2}}$$

they use two main pictures to describe this, http://aci.mta.ca/Courses/Physics/4701/EText/TimeDilation.html they are.

I am just wondering if there is another way to arrive at this equation, and if there is not, what is the...least complicated way of getting this equation, am I correct to assume it is fairly standard?


-thanks
Mark

 

[Doc Al]

That's probably the simplest way to get the "time dilation" formula. (But other (equivalent) ways exist.) Do you have a question about the derivation?

You are better off writing the equation this way:
$$\Delta t = \Delta t' \frac{1}{\sqrt{1- \frac{v^2}{c^2}}}$$
The formula tells you that an observer will measure a moving clock to be running slow compared to his own clocks. If the moving clock measures a time of Δt', the observer will measure a time of Δt.

 

[Mark]

Yes, my bad on the formula. Well essentially I'm looking for a secondary way to derive this formula (that doesn't involve any calculus past grade 12 calculus) Is it possible to do another simple derivation of this formula?

$$\Delta t = \frac{\Delta t\'\;}{\sqrt{1- \frac{v^2}{c^2}}}$$

 

[jdavel]

Mark,

Actually your formula for time dilation is a consequence of a set of equations called the Lorentz transforms, which show you how to calculate the time and location of an "event" as measured by one observer, given the time and location of the event as measured by a second observer whose moving relative to the first one. It turns out they don't get the same answer!

And good news, you don't even need to calculus to understand it! Algebra is enough. You will need an open mind (it's pretty hard to believe at first!) and a good introductory book on Special Relativity. People here can recommend one (I like the one by A.P. French, but that's because it's the one I used to learn it).

Good luck, and let us know how it goes!

 

[robphy]

Analyzing the "light clock" is the simplest approach.

No calculus is involved. It's geometry and algebra. Consult, for example,
http://landau1.phys.virginia.edu/classes/109/lectures/srelwhat.html

 

[Doc Al]

Yes, my bad on the formula. Well essentially I'm looking for a secondary way to derive this formula (that doesn't involve any calculus past grade 12 calculus) Is it possible to do another simple derivation of this formula?

Mark,
You should be well on your way by with the help given by jdavel and robphy. When I went to the link you had given, I saw the usual "light clock" diagram that robphy refers to. So I just assumed that the site gave the usual derviation of that: which involves nothing more that a little algebra (and an open mind). No calculus at all.

Let us know how you make out. As you can see, folks are eager to help here.

 

[k7swan]

Does anyone here think it might someday be possible for someone to travel at a velocity near the speed of light?

 

[Nenad]

not probable.

 

[jlh71968]

Hey Mark,

I had this similar question in my Physics III class. My copy of the time dilation equation proof is shown below as an attached zip file. The important concepts to realize are that two frames of reference are used-one being non-moving and the other moving at some velocity. In both of these frames, the speed of light (c) is constant and the distance (L) between the light source and mirror is set to some arbitrary value. By using Pythagorean's theorem, we can solve for L. Then applying some fancy Algebraic techniques, we interchange in terms of L and c because they are equal in both frames. Hope this helps.

Jay

 

[sid94]

http://www.theory.caltech.edu/people/patricia/sreltop.html

-This might turn out helpful for relativity newbies..
Hope this helps too!