Question

proof for particles of matter are always in motion and if they are then how are they moving some energy is requiered to keep moving?

Answer 1

The energy for random motion of particles of molecular and atomic level is provided by thermal energy from the surrounding. The particles absorb the thermal energy and vibrate at a very fast rate and also move from one point to another, thereby colliding with other particles and transferring energy and momentum to other particles. While in Brownian motion the suspended particles like dust are bombarded by the atoms and molecules of the fluid (gas or liquid) which results in random motion of the dust particles.