Speed in Star Trek

How fast is Impulse, Warp and Warp X speed?

As I have understood it, Warp (1) speed = 1x Speed of Light (SoL), i.e. ~300.000 km/s. However, in ST:VOY, they are hurled 70.000 LY from Earth, which will take 70 years to travel at Warp 9, i.e. Warp 9 = 1000x SoL. That makes no sense to me.

If Warp doubles for each full number (binary numbers), then Warp 10 would be 1024x SoL ~ 68 years journey. That works fine, in approximation, given they need to make various stops on the way and/or take small detours to fly through star systems on the way. But no... Not only is Warp 10 impossible (why is that, anyway?), but anything over Warp 9,0 is reserved for burst only. Thus sustained travel is limited to Warp 9, which would (by my calculations) be 512x SoL ~ 135-140 years journey. But this doesn't match any known numbers..

Bonus question: How fast is Impulse drive? You can travel in a shuttle from Earth to Jupiter in a few hours. We only ever hear about quarter, half and full Impulse speed. But at the same scale, you can travel to Jupiter in a few hours, but you can also navigate a safe landing on a planet/moon, through the atmosphere.. Again, I see no consistancy in this..

Answer

In TOS, we do not have an explicit conversion, but all stated velocities used WF^3.

The following are rounded to 1 decimal place.

So...

 OWF = Old Scale Warp Factor

 NWF = New Scale Warp Factor
   C = Speed in multiples of the speed of light
D/LY = Days per Light year 

 OWF:  1   2    3    4    5    6    7    8    9    10    11    12    13    14    15
   C:  1   8   27   64  125  216  343  512  729 1,000 1,331 1,728 2,197 2,744 3,375
D/LY: 365 45.6 13.5 5.7  2.9  1.7 1.1  0.7  0.5   0.4   0.3   0.2   0.2   0.1   0.1

TNG, DS9 and Voyager use about WF^(10/3) through warp 9.0...

 NWF:    1   2     3      4      5      6      7       8       9
   C:    1  10.1  38.9  101.6  213.7  392.5  656.1  1024.0  1516.4
 OWF:    1   2.2   3.4    4.7    6.0    7.3    8.7    10.1    11.5
D/LY:  365  36.2   9.4    3.6    1.7    0.9    0.6     0.4     0.2

Above 9.0, between 9.0 and 10, the exponent increases dramatically, and warp 10 is theoretically infinite speed.

It is worth noting, however, that in All Good Things (the closing Episode of TNG), the scale appears to continue to Factor 14 at this same WF^(10/3) progression. The on-screen stated speed for factor 13 is a match.

Further, the TNG Tech Manual mentions that this is an average speed, and that factors in local subspace topology can affect the speed in either direction.

Impulse is given several explicit speeds in various episodes. The term Full Impulse seems to be relative, and as per naval tradition, Full Impulse appears to be maximum the engines can provide, which varies from about 0.05C to an explicit 0.8C (in Voyager).

http://www.ex-astris-scientia.org/treknology/warp6.htm