The speed at which particles vibrate can vary significantly depending on various factors such as the type of particle, the medium in which they are vibrating, and the energy input into the system.
In general, particles can vibrate at speeds ranging from extremely slow vibrations, such as those in a solid structure, to extremely fast vibrations, such as those exhibited by subatomic particles.
At the atomic and molecular level, particles can vibrate at frequencies in the range of terahertz (10^12 hertz) and even higher. For example, the vibrations of atoms in a solid lattice structure typically occur in the range of several terahertz.
In the context of sound waves, particles in a medium vibrate at speeds determined by the frequency of the sound wave. In air at room temperature, sound waves travel at approximately 343 meters per second (m/s). The particles in the air vibrate back and forth, transferring energy and propagating the sound wave.
It's important to note that the speed of particle vibration does not necessarily correspond to the speed at which particles themselves move. In a wave motion, particles oscillate around their equilibrium positions without necessarily undergoing net displacement in a particular direction.
In summary, the speed at which particles can vibrate varies greatly depending on the specific circumstances and the scale at which we examine them, ranging from extremely slow vibrations in solids to very high-frequency vibrations at the atomic and subatomic level.