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.

The 10th edition of Halliday's Fundamentals of Physics, Extended building upon previous issues by offering several new features and additions. The new edition offers most accurate, extensive and varied set of assessment questions of any course management program in addition to all questions including some form of question assistance including answer specific feedback to facilitate success. The text also offers multimedia presentations (videos and animations) of much of the material that provide an alternative pathway through the material for those who struggle with reading scientific exposition.

Furthermore, the book includes math review content in both a self-study module for more in-depth review and also in just-in-time math videos for a quick refresher on a specific topic. The Halliday content is widely accepted as clear, correct, and complete. The end-of-chapters problems are without peer. The new design, which was introduced in 9e continues with 10e, making this new edition of Halliday the most accessible and reader-friendly book on the market.

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