Racetrack (game) - Mathematics and Physics

Mathematics and Physics

Each move may be represented by a vector. E.g., a move two squares to the right and four up may be represented by the vector (2,4).

The eight neighbour rule allows changing each coordinate of the vector by ±1. E.g., if the previous move was (2,4), the next one may be any of the following nine:

(1,5) (2,5) (3,5)

(1,4) (2,4) (3,4)

(1,3) (2,3) (3,3)

If each round represents 1 second and each square represents 1 metre, the vector representing each move is a velocity vector in metres per second. The four neighbour rule allows accelerations up to 1 metre per second squared, and the eight neighbours rule allows accelerations up to √2 metres per second squared. (If each square represents 10 metre instead, the size of the track and the maximum acceleration will be more realistic.)

The speed built up by acceleration can only be reduced at the same rate. This restriction reflects the inertia or momentum of the car. Note that in physics, speeding, braking, and turning right or left all are forms of "acceleration", represented by one vector. For a sports car, having the same maximum acceleration without loss of traction in all directions is not unrealistic; see Circle of forces. Note, however, that the circle of forces strictly applies to an individual tyre rather than an entire vehicle, that a slightly elongated ellipsis would be more realistic than a circle, and that the theory of traction involving this circle or ellipsis is quite simplified.