Particle acceleration

In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the particle acceleration or sound acceleration with the symbol a in metre/second². In acoustics or physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. It is thus a vector quantity with dimension length/time². In SI units, this is m/s².

To accelerate an object (air particle) is to change its velocity over a period. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation

$${\displaystyle \mathbf {a} ={\frac {d\mathbf {v} }{dt}}}$$

where

• a is the acceleration vector
• v is the velocity vector expressed in m/s
• t is time expressed in seconds.

This equation gives a the units of m/(s·s), or m/s² (read as "metres per second per second", or "metres per second squared").

An alternative equation is:

$${\displaystyle \mathbf {\bar {a}} ={\frac {\mathbf {v} -\mathbf {u} }{\Delta t}}}$$

where

• ${\displaystyle \mathbf {\bar {a}} }$ is the average acceleration (m/s²)
• ${\displaystyle \mathbf {u} }$ is the initial velocity (m/s)
• ${\displaystyle \mathbf {v} }$ is the final velocity (m/s)
• ${\displaystyle \Delta t}$ is the time interval (s)

Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have

$${\displaystyle \mathbf {a} =-{\frac {v^{2}}{r}}{\frac {\mathbf {r} }{r}}=-\omega ^{2}\mathbf {r} }$$

One common unit of acceleration is g-force, one g being the acceleration caused by the gravity of Earth.

In classical mechanics, acceleration ${\displaystyle a}$ is related to force ${\displaystyle F}$ and mass ${\displaystyle m}$ (assumed to be constant) by way of Newton's second law:

$${\displaystyle F=ma}$$