# Average Acceleration and Instantaneous Acceleration by Robert Resnick

When a particle's velocity changes from ${\stackrel{\to }{v}}_{1}$  to  ${\stackrel{\to }{v}}_{2}$ in a time interval Δt, its average acceleration ${\stackrel{\to }{a}}_{\mathrm{avg}}$ during Δt is

$\text{average acceleration}=\frac{\text{change in velocity}}{\text{time interval}}$, or

${\stackrel{\to }{a}}_{\mathrm{avg}}=\frac{{\stackrel{\to }{v}}_{2}-{\stackrel{\to }{v}}_{1}}{\Delta t}=\frac{\Delta \stackrel{\to }{v}}{\Delta t}$ (4-15)

If we shrink Δt to zero about some instant, then in the limit ${\stackrel{\to }{a}}_{\mathrm{avg}}$ approaches the instantaneous acceleration (or acceleration) $\stackrel{\to }{a}$ at that instant; that is,

$\stackrel{\to }{a}=\frac{d\stackrel{\to }{v}}{dt}$ (4-16)

If the velocity changes in either magnitude or direction (or both), the particle must have an acceleration.

We can write Eq. 4-16 in unit-vector form

$\stackrel{\to }{a}=\frac{d}{dt}\left({v}_{x}\stackrel{^}{i}+{v}_{y}\stackrel{^}{j}+{v}_{z}\stackrel{^}{k}\right)=\frac{dx}{dt}\stackrel{^}{i}+\frac{dy}{dt}\stackrel{^}{j}+\frac{dz}{dt}\stackrel{^}{k}$

We can rewrite this as

$\stackrel{\to }{a}={a}_{x}\stackrel{^}{i}+{a}_{y}\stackrel{^}{j}+{a}_{z}\stackrel{^}{k}$ (4-17)

where the scalar components of $\stackrel{\to }{a}$ are

(4-18)

To find the scalar components of $\stackrel{\to }{a}$, we differentiate the scalar components of $\stackrel{\to }{v}$.

Figure 4-6 The acceleration $\stackrel{\to }{a}$ of a particle and the scalar components of $\stackrel{\to }{a}$.

Figure 4-6 shows an acceleration vector $\stackrel{\to }{a}$ and its scalar components for a particle moving in two dimensions. Caution: When an acceleration vector is drawn, as in Fig. 4-6, it does not extend from one position to another. Rather, it shows the direction of acceleration for a particle located at its tail, and its length (representing the acceleration magnitude) can be drawn to any scale.