the answer to that is fuck yea

@Cirrus Light  
Expanding on that a bit more, I found an AWESOME resource:
 

 
The first and second stages take the thing mostly to Earth orbit - the third stage is needed to finish the job, though, for the last few hundred m/s (Earth orbit is 7,840 m/s). Getting to Earth orbit takes about 9,600 m/s, give or take a few hundred, maybe more. That 9,600 gets you to 7,840 due to gravity and aerodynamic drag losses.
 
The third stage fires again and uses all its fuel blasting the thing into a course that will overcome Earth’s gravity and send it to the Moon (Trans-Lunar Injection). This takes about 3,300 m/s.
 
As you move away from the Earth, though, its gravity slows the craft down (actually a very good thing, since it means less fuel is needed to capture into the Moon’s orbit. If you approached the moon at 3,300+7,840=~11,000 m/s, the Apollo missions would’ve been a heck of a lot harder - pretty much impossible!).
 
In a “perfect” world, you could capture into the Moon’s orbit with something like - maybe 700 m/s? But due a need for safety margins and a lot of complex geometric factors, the CSM needed to be packed with more fuel than that. It’s best to figure about 800 m/s for capture and maybe 900 for departure, just ballparking it.
 
But the real question here is what was it capable of?
 
From 28.8 tons to 11.9 tons dry with a specific impulse of 314 seconds, ^{[1] keep the LM attached the whole time since we’re not returning to Earth, we’re just blasting away as fast as we can, so add 15.2 tons [2] - and the CSM part of our insane speed run will give us 9.82ISPln(mi/mf) = 9.82314ln((28.8+15.2)/(11.9+15.2)) = ~1,500 m/s.
 
Source [2] also cites the LM as having 2,500 and 2,220 m/s capacity.
 
So add this all up:  
7,840 for Earth orbit.  
+3,300 for trans-lunar injection  
+1,500 for the CSM boost  
+2,500 for LM descent stage, and  
+2,220 for LM ascent stage.
 
17,360 m/s.
 
If we ignored gravity and aerodynamics on the initial launch,
 
17,360  
-7,840 previous launch velocity figure  
+~9,600 rough estimate of delta-vee needed to get into orbit
 
19,120 m/s.
 
19,120 / 343 = ~mach 56.
 
As for the various figures - those can be derived with the vis-viva equation., and orbital velocity is simply the balance of centrifugal acceleration (v2/r) and gravity (a = GM/r}2), yielding v = sqrt(GM/r).

@DManes  
10,000 m/s on the first stage? Something’s wrong with the math, there…
 
It takes roughly 9,600 m/s to reach earth orbit (7,840 m/s, but the extra delta-v is needed for atmospheric losses and gravity), 3,300 for trans-lunar injection, 800 for lunar orbit insertion, 2,200 for lunar descent, 2,000 for lunar ascent, then 800 again for trans-earth injection.
 
You can’t just stack all that, though, because of how the lunar orbit rendevouz architecture works. So if you launch a Saturn-V in Apollo configuration, then your best shot is to hitch a ride in the LM ascent module for maximum speed. So that would mean:
 
9,600 for SV stages I and II and a little of III.  
~3,300 for trans-lunar injection (SV stage III, known as the S-IVB).  
7-900 for lunar orbit insertion. Maybe 50-100 extra using the TEI fuel since you won’t be using it (just going as fast as you can, ignoring the “getting back to Earth alive” problem)  
Then 4,200 using the LM’s ascent and descent stages’ fuel.
 
So you can build up a good 18,000 m/s, ignoring gravity and aero drag. That’s mach 52.5, about 11 miles per second.
 
Hehehe… It’d be fun to build a Saturn V in my realism-modded KSP, then see how fast I can get a Mercury capsule going with it…
 
Anyways, you can’t feel speed in space, but you can feel acceleration, and when it takes off, various engine vibrations and aero drag effects lead to quite the ride. One of the most amazing videos ever IMO describes it very nicely!

ok lets see well its as big as a skyscraper can go to the moon and back and is the fasted thing a man can ride in oh and makes your petty sonic rainboom look weak :D RD-(eye twitches) I am going to!!  
(rocket engine starts and blast rainbow away like team rocket) cause why the fuck not!

Strap her onto it. Then let’s see if she’s skeptical.

Well, it currently holds the record for the fastest speed ever reached by a manned vehicle(39,897 km/h by CSM-106 Charlie Brown on May 26, 1969, and it also left the S-IVB and LM ascent stage in heliocentric orbit, which is quite fast), so yes.

@DManes
 
That there was a lot more homework than you really needed to do. With that said, at least you’ll pass the test on Wednesday.

Fast enough to get to the moon and back in a little over a week. Without magic.

“Yeah, but does it go fast?”
 
“Herr Dash, mein dear mare, you have no idea.”

Copied from wiki answers:
 
the maximum velocity of a rocket is approximated by:  
Delta_V = goISPln(Mi/Mf)
 
where Delta_V = the change in velocity  
go = 32.2 ft/22  
ISP = the specific impulse  
Mi = initial mass  
Mf = final mass  
mp = propellant mass  
mo = stage inert mass  
mf = stage total mass
 
This equation has to be solved for each stage considering the masses of all of the fallowing stages and payloads
 
Stage 1  
mo (lb) 300000
 
mp (lb) 4492000  
mf (lb) 4792000  
Mi (lb) 6200600  
Mf (lb) 1708600  
ISP 263  
delta V 10915.79
 
Stage 2  
mo (lb) 95000
 
mp (lb) 942000  
mf (lb) 1037000  
Mi (lb) 1408600  
Mf (lb) 466600  
ISP 421  
delta V 14977.96
 
Stage 3 mo (lb) 34000  
mp (lb) 228000  
mf (lb) 262000  
Mi (lb) 371600  
Mf (lb)143600  
ISP 421  
delta V 12889.05
 
payload mass (lb) 109600  
Max Speed ft/s 38782.80  
Max Speed Mph 26442.82
 
 
In short, ignoring aerodynamic friction and assuming a constant gravitation acceleration the maximum achievable velocity is about ~26500 mph

Why is this tagged “shuttle”?
 
/space nerd :P
 
Really though, even laying disassembled and on its side, the Saturn V screams fast.

I’ve seen the real thing up close at the Johnson Space Center in Houston, Texas. And yes, it’s HUGE!

Saturn V moon rocket? About mach 11 or so.

Those are F-1’s, the whole thing might be the base of a Saturn V. “Fast” is certainly a word you can use Rainbow. “Powerful” might be a bit more appropriate though.

RD: It needs to be about 20% faster

Damn that thing is huge.