Because the world needs good engineers and scientists, and because the kids who will grow up to become engineers and scientists need a way to get hands-on experience with physics, math and engineering.
In this age of 200-plus channels of TV, the Internet and computer games, kids are also spending far less time building tree houses, tinkering with engines, or designing downhill racers. We believe those are important skills to have. They help form the basis for good problem solving skills and an innate understanding of the real, physical world that you just can't get from a computer game, no matter how good its physics simulation software is.
Ballistic motion was one of the key players in the development of the science of physics. The word "engineer" even originated as the builders and designer of Siege Engines
Why is a budding engineering student expected to take a year or two of calculus in high school, but she isn't expected to have any real-world experience in building or working with machines and materials? Pencil and paper (or computer screens) are only one part of the learning experience. Where will she apply all of the stuff she learned in geometry and trig? Without physical projects to touch, feel and see, the lessons become abstract, their utility questionable.
A catapult project gives students a chance to see that science and engineering really can be fun, and it's a lot more than just numbers on paper. The real payoff for an engineer is in the field, where she can see and enjoy the results of her ingenuity. And it may seem counterintuitive, but engineering projects not only help kids learn math and science, they are also great at getting kids back outdoors, away from the massive over-exposure to video games, TV and the Internet.
Why all this interest in getting kids to study science and engineering? Because it's important to our society, and it's great mental cross training regardless of what field of work the kids eventually go into. Most people develop a sense for what they want to do in life while they are still in high school or even earlier. A catapult project is fun and interesting enough to inspire some kids to study the science behind how they work, and then go on to become the engineers and scientists of tomorrow.
I seem to recall building matchstick rockets, then model rockets, then Chinese rockets and learning about force, thrust, motion, weight and more. This is too cool!