Average Acceleration and Instantaneous Acceleration

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

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

${\vec{a}}_{avg} = \frac{{\vec{v}}_{2} - {\vec{v}}_{1}}{Δ t} = \frac{Δ \vec{v}}{Δ t}$ (4-15)

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

$\vec{a} = \frac{d \vec{v}}{d t}$ (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

\vec{a} = \frac{d}{d t} (v_{x} \hat{i} + v_{y} \hat{j} + v_{z} \hat{k}) = \frac{d x}{d t} \hat{i} + \frac{d y}{d t} \hat{j} + \frac{d z}{d t} \hat{k}

We can rewrite this as

$\vec{a} = a_{x} \hat{i} + a_{y} \hat{j} + a_{z} \hat{k}$ (4-17)

where the scalar components of $\vec{a}$ are

$a_{x} = \frac{d v_{x}}{d t}, a_{y} = \frac{d v_{y}}{d t}, and a_{z} = \frac{d v_{z}}{d t}$ (4-18)

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

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

Figure 4-6 shows an acceleration vector $\vec{a}$ 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.

About the Authors

David Halliday was an American physicist known for his physics textbooks, Physics and Fundamentals of Physics, which he wrote with Robert Resnick. Both textbooks have been in continuous use since 1960 and are available in more than 47 languages.

Robert Resnick was a physics educator and author of physics textbooks. He was born in Baltimore, Maryland on January 11, 1923 and graduated from the Baltimore City College high school in 1939. He received his B.A. in 1943 and his Ph.D. in 1949, both in physics from Johns Hopkins University.

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