Question 3.3: CALCULATING PARALLEL AND PERPENDICULAR COMPONENTS OF ACCELER...

IDENTIFY and SET UP:

We want to find the components of the acceleration vector \overrightarrow{\boldsymbol{a}} that are parallel and perpendicular to velocity vector \overrightarrow{\boldsymbol{v}}. We found the directions of \overrightarrow{\boldsymbol{v}} and \overrightarrow{\boldsymbol{a}} in Examples 3.1 and 3.2, respectively; Fig. 3.9 shows the results. From these directions we can find the angle between the two vectors and the components of \overrightarrow{\boldsymbol{a}} with respect to the direction of \overrightarrow{\boldsymbol{v}}.

EXECUTE:

From Example 3.2, at t = 2.0 s the particle has an acceleration of magnitude 0.58 m/s^2 at an angle of 149° with respect to the positive x-axis. In Example 3.1 we found that at this time the velocity vector is at an angle of 128° with respect to the positive x-axis. The angle between \overrightarrow{\boldsymbol{a}} and \overrightarrow{\boldsymbol{v}} is therefore 149° – 128° = 21° (Fig. 3.13). Hence the components of acceleration parallel and perpendicular to \overrightarrow{\boldsymbol{v}} are

EVALUATE: The parallel component a_{\|} is positive (in the same direction as \overrightarrow{\boldsymbol{v}}), which means that the speed is increasing at this instant. The value a_{\|}=+0.54 \mathrm{~m} / \mathrm{s}^{2} tells us that the speed is increasing at this instant at a rate of 0.54 m/s per second. The perpendicular component a_{\perp} is not zero, which means that at this instant the rover is turning—that is, it is changing direction and following a curved path.