When you start talking about trajectories, people automatically think about rockets. They talk about having to plot the trajectory of a rocket but what other things can we bring to the discussion on this topic.
The starting point would be to explore the multiple meanings of trajectory because it is easy to confuse at least two of the meanings. I tried looking under trajectories and learning trajectories and could not find exactly what I wanted. It wasn't until I looked at Projectile Trajectories that I found exactly what I wanted.
So how do you use projectile motion to teach math? Well in most cases, even with bullets, the path is all or part of a parabola. The Physics Classroom has a wonderful list of interactive simulations that can be used in the classroom starting with its own simulator. Their simulator allows you to trace the parabolic path of the projectile, read the x and y displacement, and the velocity vectors. It is easy to stop and start the path so a student can find the max height and displacement so they can calculate the formula for the path. In addition, the speed, starting height, and launch angle can be adjusted according to needs. Finally, this activity comes with a 4 page worksheet that turns this activity into a real exploration.
Out where I live, we have tons of hunters. It would be easy to teach a lesson on bullet trajectories and possibly ways of finding the equations. I do not know anything about using guns at all so I'm off to the internet to find information. Well it turns out there is a site that has quite a lot of information including this site which has information on the Mathematics for Precision Shooters. I never realized there were so many different formulas involved in shooting. I thought you just aimed and shot. The site is for serious shooters who may be in a competition or perhaps in the military. It is mind boggling.
The last item to look at is chucking a Pumpkin which is serious business for some people. This site has an article where the author discusses how he came up with his values in order to calculate the height of a pumpkin in a toss. I love the way the author goes through each and every value. It gives the students a chance of understanding everything involved. Check it out.