Click on green flag to see trajectory determined by the slider settings. Change either or both settings by 1 and observe that the output (where the projectile lands) is close to the previous shot. "Linearity' in this sense means outputs are proportional and predictable from inputs. The purpose of this project is to demonstrate the ‘small change in input’ creates a ‘small change in output’ that is characteristic of linear systems.
The equation of motion for a projectile (like a cannonball) is, Y =(-g/(2vx0)^2)x^2 + (vy0/vx0)x where vx0 is the x-component of the initial velocity v, vy0 is the y-component of the initial velocity, and g is the acceleration of gravity. This is a quadratic equation of the form y = ax^2 + bx + c with the coefficients a and b as shown in the equation. Even though the equation graphs as a parabola, projectile motion is considered to be a linear system (not to be confused with y = mx + b). It’s linear because a small change in the input(s) initial velocity and/or launch angle produces a small change in the output, that is, where the projectile lands. Deterministic chaos is the study of nonlinear systems. That is, systems where a small change in input can cause a huge change in output. In chaos theory, that is the basis for the Butterfly Effect (see my Lorenz Attractor project).
