Excellent idea!  As the slingstone experiances constant deceleration the more sheets of newspaper the better but old newspaper is cheap.

Freshmen Physics textbooks typically have ballistics problems.   The problems are based on the following truths:

Energy is always conserved.
Kinetic-Energy is not always conserved.
Momentum IS ALWAYS conserved.

The speed of a projectile can be very precisely measured using a pendulum a ruler and a weight scale.

The setup:
The pendulum should have a 'bob' on the end that the projectile will sink into.  It must not bounce off.  The porjectile must imbed inside the bob for the following calculations to work. The 'bob' of the pendulum is the target.

Use the ruler to measure how high the pendulum swings after being hit. Use the scale to measure the mass of the 'bob' and the projectile.

Then consult your physics book to see how they work out the problem. Basically it works like this:

The momentum of the projectile before it strikes the 'bob' is equal to the momentum of the 'bob' with the projectile in it right after impact.  That momentum causes the kinetic energy to be transferred to the 'bob', wich in turn gets converted to potential energy as the bob starts to swing up in an arc.  I'll post a final solution later. This method can be VERY accurate.  The accuracy can be as good as your scale for measuring the bob's mass and the accuracey for measuring how high the pedulum swings.

A back of the envelope calculations would yield the following:
Givens:
projectile = 100 grams
bob swings 33 cm upward
bob =3900 grams

Answer
Velocity of projectile = 100 m/s.

ArizonaSlinger, I was thinking the exact same thing just a few days ago.  I was thinking shooting a blob of clay at the target/pendulum, which would just be a weighted box.  However I decided it would be too hard to do, especially getting the swing height measured accuratly (and because it would take me many tries to hit the box).  Maybe a witness or video camera would help. 

Wanderer - That's a good idea!  As long as you used very thin targets like you said, I think it would work well!  Some large sheets of tissue paper would probably be best, but news paper would also work.  Microsoft sound recorder, which all window users have, lets you time sounds down to the 1/100th of a second, so with that you could easily do it.  You could go through it pausing at the bursts of sound that appear on the line, and note the times.  Also i'm guessing that ten as opposed to five would maybe be better, because it would be easier to tell the two sounds apart on the recording.

I think most measureing systems would have some built in inaccuracies, with the pendulum some energy would be lost to heat,  think of fire pistons, with paper energy spent on breaking the barriers but probably a tiny percentage.  I like the ease of set up for paper.  Maybe sound could be used another way too.  What if one made two recordings, one through a mike behind the slinger and one through a mike behind the target.  Use a sling that cracks and sling a buzzer.  Load both recordings into a computer, use the crack  and sound of impact to line them up on the screen and compare doppler effects.  It may be posable to calculate the velocity and deceleration  based on the differance in  pitch between the two recordings..

I've been mulling this one over and thought of a frill.  If one synchronized two watches and put  one by each mike beeping occasionally at the same times ( or one watch in the middle if one was sure that one wouldn't hit it on a unlucky  bad shot)   it would help to line things up back at the pc and serve as  check on the playback and recording speeds being the same.  The check would matter more for tape based machines, which may be more accurate as they are analogue but tend to slow as the battery drains.  it will end up chopped up into digital on the pc anyway.  This form of measurement would also waste energy of course, buzzers only buzz because they are irregular rocks and so a bit slower.  The sound as they rotate through the air is probably irregular too, maybe "comparing individual waves" would have been better than saying pitch.  Still it would be easy, just set it up and apart from scaring passers by shouting things like "Shot 32, apache!!" one could just be slinging normally.

I found two old threads About making humming or whistling projectiles.  I've been meaning to try slinging iron nuts myself to see how well they hum.  Both of the threads have some decent info, but neither is very long.

http://www.slinging.org/forum2/yabb/YaBB.cgi?board=1;action=display;num=11299135...

http://www.slinging.org/forum2/yabb/YaBB.cgi?board=1;action=display;num=11452740...

However, as much fun as freshman engineering projects can be, if I ever clock my speed, I'll stick with the two sheets of paper and a microphone  .

If I did have a gold glans I'd give it to some mathematician to sort this out, I've been trying to work out the method for converting the  data into information and it's driving me mad, any physicists please help.  The experiment would be easy enough to set up, though I'd have to cadge a second recorder from somewhere ( if I mention that  mine would be the one in the "target bunker" that should be ok).  As I live inland and not near any handy mature rivers almost all the rocks I sling travel loudly ( you're a fortunate man Wanderer ) but my guess is that the sound will be a bit inconsistent, no idea if that is a good or a bad thing, so something more regular may be helpful.  I've just read those threads ( cheers Cliff )  and the one about the singing arrow gives me an idea.  I don't have a masonry drill but maybe molding glans  around sections of drinking straws would give the same effect, try to work out where they go with a bit of plasticine first before wasting high density concrete,  worth a try anyway.

The experiment is not too much of a problem,  but probably won't happen any time soon  as I only have the one afternoon a week and it's cordage at the moment, plus it would be nice to actually sling a bit plus weather, my bottom of the range mini disc can't handle damp.  My lack of knowledge is a problem.  This is where I could use some help.  So .....factors which will be known and available tools..
1/  The distance between target and slinger
2/ The speed of sound, which varies far more in air than I would have guessed,  is no problem as I found this http://www.sengpielaudio.com/calculator-speedsound.htm ; enter temperature and it does it for you, a link of the same page has another for humidity and pressure.  I can measure temperature and just hope that the local met report is close enough for the rest.
3/ I found a sound editor http://audacity.sourceforge.net/ ;  One can display sound as a waveform with a  time line underneath it.  One can zoom in on the wave/time line  to pretty high resolution.  Easily enough I think.

Lets imagine it's done and on my machine.  I'm going to pretend the distance from the mics to the target/slinger don't exist as I have enough problems without trig', it'll only be half a meter or so anyway.  On the assumption that the speed of sound is 340 meters per second on the day and that I sling 34 meters ( I'm lazy that way ).   I would expect the sound of the crack to be live at the slingers end and  delayed by a tenth of a second at the targets, and visa versa for the sound of impact.  I would also expect the Doppler effect, the waves would arrive closer together for the target microphone and further apart for the slinger microphone. does this mean the the slingers end recording will be slightly longer?   I have read that the Doppler effect does not change the speed of sound.  So if I zoom in on a distinctive pointy bit ( forgive the technical term from now on I'll call it a “dpb” ) and note how long it is after the crack and before the impact, then do the same with it's twin dpb.  How do I turn this into location?

Oh drat, just realized I may have spent hours on the wrong thought experiment.  'o' level geology!  Put the microphones either side of slinger,  quite a long way apart.  Put a timepiece in the middle beeping at regular intervals, or one with each.  The time the sound of the dpb's take to reach the microphones will give the distance to the glans, then it is just drawing two circles.  When one detects earth quakes one needs a third in order to tell which crossing point is the epicenter but hopefully I won't miss by 180 degrees. If the glans is really loud one could even forget about targets all together and just sling as hard as one can in the right general direction, the microphones would track the glans for as long as they could hear them. Unless the Doppler effect messes it up, which depends on how fast the glans retreats .....which is what we're trying to find out.... because ....we ...don't... know!   Unless there's some way round that...we have the beeps.  Or would it just be timeing when dpb's happen without knowing where?  Third mic after all?  Oh I don't know.

Arggghhh it's another one of those itchy ideas.  If anyone cares to throw some knowledge,ideas,or  just do it properly I'd be grateful, this one has kept me from sleeping most of the night. I have that  sort of mind, once it was “I wonder what the surface area of  the earth is in light years?” this is worse.   The wiki writing people want the velocity info too.