RP Rules: Official RP Warp Calculator

[Evshell]

Please use this calculator for all warp speed references in the RP:

http://convertalot.com/warp_factor_calculator.html

Make sure to click "The Next Generation" which uses the Warp 10 speed limit and standard formula that I have seen and used heavily before, and most accurately estimates warp speeds, including the special calculation between warp 9 and 10.

With transwarp, simply take the transwarp factor to the 9th power to determine how many C (speed of light) you are traveling. So for example, transwarp 3 is the speed of light times 3^9 or 19,683C.

Using a distance D, and the warp speed W, then calculate D/W to determine how many years it will take you to reach the distance at that speed. For example, at transwarp 3, from Earth to Alpha Centauri (4.3 LY away) it would take 4.3 / 19683 = 0.00021846263275 years.

In order to figure out how many days that is, multiply by 365, or 0.07973886 days.

For hours, multiply again by 24, or 1.91373 hours.

Interesting "Fact" #1:
To reach the Andromeda Galaxy at transwarp 3 it would take 101.6 years. To reach it at transwarp 5 (maintained the whole way) it would take a little over ONE YEAR! This is because transwarp 5 calculates to 1,953,125C and the Andromeda Galaxy is approximately 2 million LY away. Using the calculation mentioned above of D/W, you should see that this comes out to almost exactly one year.
[Edit] It appears I was wrong on this. Although 1 year sounds nice and round and awesome, it is actually 1.26 years, due to the fact that the Andromeda Galaxy is ACTUALLY 2.5 million LY away. The 2 million figure was from the warp speed calculator link, but real science says it is 2.5 million.

Interesting "Fact" #2:
Even at the lowest super-warp transwarp rate of 3, the USS Horizon and other future transwarp-capable ships could reach the Andromeda Galaxy three times as fast as the fastest rate previously achieved by the Kelvan race. At a safe and consistent transwarp rate of 4.5, it is a 2.6-year journey, which is 115 times as fast as the Kelvans.

[Evshell]

With the previous thought of Transwarp limit in our RP being 5, I didn't think it appropriate to confuse things and bring this up, however, with recent events it may be warranted now.

If you review the calculations presented in the provided warp factor calculator link, you will notice a special function that is added to the power of 3.3333 that is applied when warp factor is between 9 and 10. This was added to explain the theoretical "warp speed limit" reached by Tom Paris in the dreaded Voyager episode that I shall not name.

The same function is needed when dealing with Transwarp. Although the Transwarp speeds achieved at 3 and above are orders of magnitude huger than Warp 9, it should be noted that both Warp and Transwarp will result in speeds approaching infinity when either factor approaches 10. If you are at all familiar with limits from High School or College calculus classes, you may note the following:

The lim(W) as V approaches infinity is 10, where V is velocity and W is either Warp OR Transwarp.

This calculation indicates that you may never go infinite velocity, but that in trying so you will approach, but not quite reach Warp 10 or Transwarp 10. Of course, Tom Paris proved the calculation slightly wrong, but in doing so, received almost deadly consequences. So let's just forget about that episode, and say "never", shall we?

So you may be asking me, "What's the difference, then, between Warp and Transwarp, if they end up at the same place?"

Well you may have heard the saying "It's not the destination that matters, it's the journey." This saying is quite apt in this case. Transwarp may reach the same destination as Warp at a factor of 10, however, it gets there a lot faster a lot sooner. As you can see from the attached graph, Transwarp reaches higher speeds (goes further up) as you approach factor 9. This is represented in a logarithmic graph. Mathematicians use logarithmic graphs to compare exponential increases when the real graph would be too astronomical to view.

Note: For reference, the 8 in the velocity axis of the graph represents going approximately 134 million times the speed of light, whereas the 3 represents approximately 1250C. Also, I screwed up a little bit and the 0 should really be 1C, or the speed of light, but you get the picture.

You can then see from the far right side of the graph, that Warp reaches factor 9 at a much slower velocity than Transwarp, but then they both increase drastically and shoot upwards toward infinity, and will not cross the threshold of factor 10.

Hope this explains a bit more about the differences and why, in the recent Horizon post, Kheren mentioned the ill-fated journey of Tom Paris as he reached Transwarp 10. Many of you might've been saying, "but he only reached Warp 10." This was not a mistake on Kheren's part, as those two factors are identical at that point.


Note: If you would like to try out and experiment with the formulas for yourself, go to: https://www.desmos.com/calculator and copy and past the following two formulas in exactly as shown (if you want to fix my error, just put "1+" at the beginning of the formula (after the = sign):

Warp
$y=\log \left(x^{\left(\frac{10}{3}-\left(.5\cdot \log \left(10-x\right)\right)\right)}\right)$

Transwarp
$y=\log \left(x^{\left(9-\left(.5\cdot \log \left(10-x\right)\right)\right)}\right)$