## 2-4: Acceleration.

when the velocity of a particle changes with time, the particle is said to be accelerating.

– The Average acceleration  $a_{x,avg}$  of the particle is definded as the change in velocity  $\Delta v_{x}$  divided by the time interval  $\Delta t$  during which that change occurs:

$a_{x,avg}\equiv \frac{\Delta v_{x}}{\Delta t}=\frac{v_{xf} - v_{xi}}{t_{f}-t_{i}}$

Instantaneous acceleration: it is useful to define it as the limit of the average acceleration as  $\Delta t$  approaches zero.

$a_{x}\equiv \lim_{\Delta t\rightarrow 0}\frac{\Delta v_{x}}{\Delta t}=\frac{dv_{x}}{dt}$

the instantaneous acceleration equals the derivative of the velocity  with respect to time

$a_{x}=\frac{dv_{x}}{dt}=\frac{d}{dt}(\frac{dx}{dt})=\frac{d^{2}x}{dt^{2}}$

that is, in one-dimensional motion, the acceleration equals the second derivative of  $x$  with respect to time.

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