Can I build a TNT cannon that fires the TNT directly upwards?

I've successfully built a cannon that will fire the player upwards to above the clouds by using this design (found on YouTube). The player stands on the stone pad, it triggers the TNT and the "water compressor" pushes the TNT towards the player and detonates, sending the player flying.

I've tried replacing the stone plate with a half-slab and dropping activated TNT onto it but it doesn't work - the TNT launcher detonates but the "ammo" doesn't go anywhere and then blows up, destorying torches and redstone for the activiation circuit underneath.

Is this possible to do?

Yes

First, build the base, add water:

Add a holder for the 'firework' tnt block one block above the center:

Wire it up. Two repeaters at full spacing will give a good delay:

Test at this point, make sure all the wires light up when you hit the button. If any fail to, you'll blow it all up. Next, load it!

5..4..3..2..1...

BOOM!!!

When activated, TNT jumps to the side a little bit, so in order to make sure the ammo doesn't hit the edge of the 1×1 hole in the top layer, you should build a tube around the ammo (one block above the slab) to make sure it sees a smooth "barrel" rather than hitting the edge. You can also protect against accidents by placing water in this tube (though there will be some blast leakage anyway).