Session 3

Real vs Hyper-Real 2d Newtonian Mechanics

Recall that standard 2d Newtonian Mechanics with constant velocity (vx, vy) in (x,y)-direction is described by the equations for particle paths:

  • dx=vx*dt, dy=vy*dt.

Particles follow straight lines with x(t) = x0 + t*vx and y(t) = y0 + t*vy, where (x0,y0) denotes position for t=0. 

Continue to recall that standard 2d Newtonian Mechanics with constant acceleration (ax, ay) is described by

  • dx=vx*dt, dy=vy*dt,
  • dvx=ax*dt, dvy=ay*dt.

The acceleration can be (0,1) given by constant gravitation in the y-direction (down). Particles follow parabolas as 2nd-degree polynomials (AngryBirds): If ax = 0, ay = 1, vx0 = 1 and vy0 = 0, then x(t) = t, vy(t) = t and y(t) = t*t/2, that is y = x*x/2 as the equation of a parabola.

Consider a generalisation to

  • dx=vx*dt, dy=vy*dt,
  • dvx=ax*dt, dvy=ay*dt
  • dax=jx*dt, day=jy*dt

with constant jerk (jx,jy) where particles follow 3-rd degree polynomials. Compare standard 2d Newtonian mechanics with constant acceleration (0,1), with hyper-real 2d Newtonian mechanics with constant jerk (0,1). Generalise.

Compare with this example. Compare with Games 3 + 4. Extend to control by jerk.