After many frustrating trial and error attempts, we finally decided to resort to mathematics as a way of determining what the requirements would need to be to shoot the particle balls into the center goal.

Projectile motion was first analyzed by Galileo in 1638. He showed that a projectile will follow a parabolic path, determined by the initial launch conditions and the acceleration of gravity, and independent of the mass of the projectile. This sketch shows the conditions for our particle launcher on the robot:

The maximum trajectory height value is important because Rule RG08 states that the particle cannot rise more than 72 inches above the floor. (It isn’t clear in the game rules what will happen if a robot exceeds this number, however. There is no specified penalty.)

Applying the above analysis, we can construct a table using Excel that explores the effect of various launch conditions:

There are several interesting conclusions we can draw from this table:

- The steeper the launch angle, the greater the possible variation in launch velocity to achieve the target.
- Launch angles of less than 58 degrees won’t work at all: a velocity large enough to reach the leading edge of the target will cause the particle to overshoot.
- Launch angles of greater than 74 degrees will violate the 72″ trajectory maximum height rule.
- So the optimum launch angle is about 70 degrees to maximize the velocity tolerance while providing some margin to avoid too high a trajectory. This will allow about a 10% variation in velocity with a maximum trajectory height of about 62″.