4.1.2 Acceleration Mathematically

In Tactics Box 4.1, the average acceleration is found from two velocity vectors separated by the time interval $Δ t$ . If we let $Δ t$ get smaller and smaller, the two velocity vectors get closer and closer. In the limit $Δ t \to 0$ , we have the instantaneous acceleration $\vec{a}$ at the same point on the trajectory (and the same instant of time) as the instantaneous velocity $\vec{v}$ . This is shown in FIGURE 4.6.

The figure labeled ("a") illustrates the parallel and perpendicular components of instantaneous acceleration of a particle.

The figure labeled (b) illustrates the x- and y- components of instantaneous acceleration of a particle.

Full Alternative Text

By definition, the acceleration vector $\vec{a}$ is the rate at which the velocity $\vec{v}$ is changing at that instant. To show this, FIGURE 4.6a decomposes $\vec{a}$ into components ${\vec{a}}_{∥}$ and $\vec{a}_{⊥}$ that are parallel and perpendicular to the trajectory. As we just showed, ${\vec{a}}_{∥}$ is associated with a change of speed, and $\vec{a}_{⊥}$ is associated with a change of direction. Both kinds of changes are accelerations. Notice that $\vec{a}_{⊥}$ always points toward the “inside” of the curve because that is the direction in which $\vec{v}$ is changing.

Although the parallel and perpendicular components of $\vec{a}$ convey important ideas about acceleration, it’s often more practical to write $\vec{a}$ in terms of the x- and y-components shown in FIGURE 4.6b. Because $\vec{v} = v_{x} î + v_{y} ĵ$ , we find

\vec{a} = a_{x} î + a_{y} ĵ = \frac{d \vec{v}}{d t} = \frac{d v_{x}}{d t} î + \frac{d v_{y}}{d t} ĵ

(4.9)

from which we see that

a_{x} = \frac{d v_{x}}{d t} and a_{y} = \frac{d v_{y}}{d t}

(4.10)

That is, the x-component of $\vec{a}$ is the rate $d v_{x} / d t$ at which the x-component of velocity is changing.

NOTE Figures 4.6a and Figure 4.6b show the same acceleration vector; all that differs is how we’ve chosen to decompose it. For motion with constant acceleration, which includes projectile motion, the decomposition into x- and y-components is most convenient. But we’ll find that the parallel and perpendicular components are especially suited to an analysis of circular motion.