What coordinate systems do you prefer for the study of motion of a particle in a two - dimensional plane? How about for the motion of a particle in a three-dimensional space?

1) For motion in two dimensions Cartesian co-ordinate system is generally preferred though polar co-ordinate system can be also used for problems involving circular motion.

2) For a general plane motion we can choose any frame depending on the problem that makes it easier to analyse but in general Cartesian coordinate system is preferred.

Explanation:

Motion of particles in space or in planes can be arbitrary in nature meaning that the particle can choose any path to travel. So for the cases of planar motion the Cartesian co-ordinate system is preferred as we are to deal with only 2 unknown position co-ordinates of the particle. But if the motion is circular such as around a circle then the analysis is much simplified by using polar co-ordinate system.

Similarly depending on the motion of the particle in space we can choose Cartesian or spherical or cylindrical co-ordinate system for easier analysis depending upon the nature of the problem.