Acceleration
In addition to velocity, which is the rate of change of the position, the description of the motion often requires us to be able to quantify how fast the velocity itself is changing. The quantity that does this is called acceleration.
For example, we are often told in what time a given car can reach a speed of 100 km/h. What we can obtain from this information is the average acceleration; in a way similar to what we have done with instantaneous velocity, we can refine the time interval to arrive at the concept of instantaneous acceleration.
Acceleration
Average acceleration is the change in velocity \(\Delta v_x\) divided by the time interval \(\Delta t\) during which that change occurs:
\[\overline{a_x} := \frac{\Delta v_x}{\Delta t} = \frac{v_{x} - v_{x0}}{t - t_0}.\]
Instantaneous acceleration is the average acceleration calculated in the limit as \(\Delta t\) approaches zero:
\[a_x := \lim\limits_{\Delta t \to 0}\frac{\Delta v_x}{\Delta t}.\]
This is, by definition, the first derivative of the velocity with respect to time, which is – since the velocity is the first derivative of the position with respect to time – equal to the second derivative of the position with respect to time:
\[a_x := \frac{\mathrm{d}v_x}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{\mathrm{d}x}{\mathrm{d}t}\right) = \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}.\]
Acceleration is a vector quantity. The direction of the acceleration is determined by the direction of the change in the velocity (see the figure below).
The SI unit of acceleration is the metre per second squared:
\[\left[a_x\right] = 1\,\frac{\textrm{m}}{\textrm{s}^2}.\]
