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Check Your Understanding Suppose the acceleration function has the form $\overrightarrow{a}(t)=a\widehat{i}+b\widehat{j}+c\widehat{k}\text{m/}{\text{s}}^{2},$ where a, b, and c are constants. What can be said about the functional form of the velocity function?
The acceleration vector is constant and doesn’t change with time. If a, b , and c are not zero, then the velocity function must be linear in time. We have $\overrightarrow{v}(t)={\displaystyle \int \overrightarrow{a}dt=}{\displaystyle \int (a\widehat{i}+b\widehat{j}+c\widehat{k})dt=}(a\widehat{i}+b\widehat{j}+c\widehat{k})t\phantom{\rule{0.2em}{0ex}}\text{m/s},$ since taking the derivative of the velocity function produces $\overrightarrow{a}(t).$ If any of the components of the acceleration are zero, then that component of the velocity would be a constant.
Multidimensional motion with constant acceleration can be treated the same way as shown in the previous chapter for one-dimensional motion. Earlier we showed that three-dimensional motion is equivalent to three one-dimensional motions, each along an axis perpendicular to the others. To develop the relevant equations in each direction, let’s consider the two-dimensional problem of a particle moving in the xy plane with constant acceleration, ignoring the z -component for the moment. The acceleration vector is
Each component of the motion has a separate set of equations similar to [link] – [link] of the previous chapter on one-dimensional motion. We show only the equations for position and velocity in the x - and y -directions. A similar set of kinematic equations could be written for motion in the z -direction:
Here the subscript 0 denotes the initial position or velocity. [link] to [link] can be substituted into [link] and [link] without the z -component to obtain the position vector and velocity vector as a function of time in two dimensions:
The following example illustrates a practical use of the kinematic equations in two dimensions.
and
(a) What are the x- and y -components of the skier’s position and velocity as functions of time? (b) What are her position and velocity at t = 10.0 s?
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