Do you do handstand pops/jumps/hops? (There are many different names for them!)
If not - here's how to do them:
1. You must have a really good fast handstand - Start standing with feet together, arms down at your side, or even a little behind you. Take a large step forwards, chest down. Swing your arms to your ears and reach forwards as far as you can before putting your hands down whilst at the same time kicking quickly but strongly to handstand.
2. In press up postition, shrug your shoulders up and down without bending your arms - do about 30 of these. As they get easier, raise your feet until eventually you are in handstand against a wall.
3. Shoulder shrugs in press up position - on the 'up' phase, actually lift your hands off the floor, but be sure not to bend your arms.
4. Put these exercises together: Start with number 1. As you kick to handstand with the first leg, shrug your shoulders and jump forwards and upwards, finishing in a handstand with legs glued together and head in neutral.
If you can, get your coach to support you - as the first leg reaches handstand, ask your coach to actually lift you at the hips. The fast joining of the legs should help though as it will create a weightless sensation.
Make sure you enter the handstand with your shoulders shrugged up.
The actual 'pop' off the vault however is slightly more difficult as your legs are already together. It is a matter of hitting the vault at the correct angle. You should hit the vault just before handstand, with your shoulders shrugged up. The 'pop' happens as you reach vertical and you should have a good 2nd flight.
The handstand jumps really help though. Once you get really good at them, try jumping up a panel mat, then making a series of steps to try and jump up! Its tough, but really useful for the 2nd flight!
Good Luck!
Marie