Time Dilation, Again - Part 2 in 26th Dimension Series

PART 2  - Time Dilation, again

Imagination is everything. It is the preview of life's coming attractions - Albert Einstein 

 

Last time, we used a formula that we did not derive ourselves, so let’s derive the formula for the Time dilation in an easy way.

Pic credit- Encyclopedia Britannica, inc. 

So, let’s do the assumptions, there’s VBT and the adorable pet Calus hanging around. On their way back to  their home, they saw a boy playing with a ball on a moving train. Let

 

 

Let’s assume for that boy on the train, the phenomenon seemed like the one that is shown in the drawing above. So, it covers a distance of 2L and then returns to point A.

The related equations are as under:

And now, we need a formula for L , which is as under:

We will use this formula at the end of this derivation. Now, you might ask, why formula for L? The answer is, we have to get rid of L because we don’t want L in our formula. We just need variables of time and speed.

 

Now is the turn for VBT and Calus. They saw something different. I will just leave a drawing below:

In this case, the ball covered a distance equal to the length of the curve PQR, but to calculate that we need integral calculus. So, to simplify things, we consider it a V-shaped path, as traced by the blue-dotted line. So the distance covered by the ball in this case will be twice the h.

Now, we need to calculate value for h. So, let’s focus on the right-angle triangle PBQ, where h is given by the Pythagoras Theorem as follows:

But, we don’t know the value of b, but remember, we said the train is moving with a velocity of

 

Putting back this value into our previous formula, we get,

\To avoid any pain, we will simplify h and this becomes, 

 

Now, The formula for t_2 is as under:

After putting the value of h, we get,

Now, to get rid of L term, we put its value, and after putting its value, we get

Now, check out the last steps of the derivation:

After this step, we take square root on both sides, and then, boom!

As said, we just derived the formula for the time dilation in one of the easiest ways possible. But still this is not the end of the story, in the next article, we will discuss this further, until then, stay tuned for 26^th Dimension of the Universe.