Found a YouTube clip of someone
using the Audacity sound recording method to measure the speed of a slingshot. The film clip maker uses a "double screen" method with a distance of only 2 metres between them and shoots so the projectile passes through both targets. For an actual sling I think the double screen method is too complicated as it requires quite accurate shooting to hit both screens. I found it much easier to just shoot at my garage wall from a distance of 8 metres and simply record the swishing sound of the release and the whack of the ball hitting the garage wall. I would suggest a minimum of 5 metres from firing line to target as the ball may have already moved some distance towards the target when the "swish" of the release occurs, so too short a distance may introduce an error.
He also places the recording microphone midway between the slingshot and the target. For slinging you may want to move the microphone/laptop further back if there is any risk of it being damaged. Even if it is not equidistant from both the error will be very small.
So the way you record, the distance and times you get will vary from those on the video but the actual data analysis method is the same and clearly explained.
BTW I have also used the same single target method to measure blowgun speed with very satisfactory results. It would probably even work for airguns if set up carefully.
(Ironically for a method involving sound I suggest watching the clip with the sound off as the background 'alleged music' is atrocious.)
http://www.youtube.com/watch?v=fjeQbDMzAak