There are several ways to do this. One is measuring the horizontal distance the proyectile reaches when you launch your catapult on a horizontal surface. Then you can use proyectile equations...however you will need to know the initial velocity.
It really depends on your catapult. You could paste a protractor to your catapult and see the angle it reaches at the moment it releases the proyectiles, or if you want to do it more accurately, you could record it, so that later, using slowmotion you can see exactly the angle, using the protractor as reference. Other ways include measuring the time of flight and the horizontal distance to find horizontal velocity.
Lol, let's see what other forumers have to say.