Acceleration

Acceleration is defined as the rate of change of velocity. The units of measurements for acceleration can be implied from the definition to be meters/second divided by seconds, usually written $$m/s^2$$.

Acceleration is inherently a vector quantity, and an object will have non-zero acceleration if its speed and/or direction is changing.

Average acceleration

The average acceleration is given by:

$\vec{a}_{average}=\dfrac{\Delta\vec{v}}{\Delta t}=\dfrac{\vec{v}_2-\vec{v}_1}{t_2-t_1}$

where the small arrows indicate the vector quantities. The operation of subtracting the initial from the final velocity must be done by vector additionsince they are inherently vectors.

Instantaneous acceleration

The instantaneous acceleration at any time may be obtained by taking the limit of the average acceleration as the time interval approaches zero. This is the derivative of the velocity with respect to time:

$\vec{a}_{instantaneous}=\lim_{\Delta t\rightarrow 0}\dfrac{\Delta\vec{v}}{\Delta t}=\dfrac{d\vec{v}}{dt}$