Back in 1964 our High School physics class tried an experiment.  It fell in line with a recent MythBuster's episode, "Super Sonic Ping Pong Balls.

We shot ping pong balls at a high velocity ( no means to measure speed then ) but they bounced off the back wall of our rather sizable auditorium.  Old high schools had huge auditoriums and ours had a balcony and probably sat 1000 people to give you a mental picture of the distance from the launch site on the stage to the upper back wall.  Probably over 100 foot distant.

We did not use air and no barrel to speak of.  We used a plasma arc as the method of propulsion.

I don't recall the original design, that was WAY BACK in 1964 in Ohio.
But the short version anyone with a little electronics can understand is that we discharged a rather hefty current through a very fine wire located just under the ball.  POW, it sounded like a shotgun and the ball seemed to vanish.  Then the click of the ball bouncing from the rear of the auditorium.

Here is a possible method to reproduce this. Caution is advised as the power being used could be dangerous, even lethal under the wrong circumstances.
The launcher could be little more than a PVC fitting sized right to hold the ball.  Drill two small holes across from each other to lie just under the ball.  Put a pvc plug or cap in the other end, possible with a screw through it to hold it to a table.

Push two small wires thru each hole ( make the diameter right for the wire )  Strip a short length of finely stranded wire for your ignition source and attach it between the two larger wires so the fine wire is perhaps 1/2" long and just under the ball. ( feel free to experiment with length and method to attach. You could solder or try wrapping the fine wire around the other wires.

For the power source, check your local photo processor or drug store who accepts throw away film cameras.  Often they will give you the empties.  Ask for some with a flash.

Start small.  Use the flash as it came from the camera. ( Always make sure to discharge the capacitor !! )  Unsolder or remove the flash tube from the circuit and where the two leads connected to the tube add two wires just long enough to reach the two wires in your launcher.

There are more sophisticated methods to fire this thing but if you can find one of those old knife switches that and a piece of string should provide you with a stand-off trigger.  Put that in series with the capacitor and the launcher.

Add batteries and let the flash circuit charge until the indicator light is on.

Using a string or a stick or some method to keep your hands away from everything, slap that knife switch closed.  There should be a POW and the ball should be airborne.

For more power ( distance and speed ) you will need to replace the existing capacitor with a larger one or add more to the existing in parallel.  Make sure you ALWAYS discharge these capacitors before you touch them.  You could get a nasty shock if you don't.

Let me know how it goes.