Objectives
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Understand the components of the acceleration vector and how the components affect the speed and direction of the object.
Section 7.8 Splitting the Acceleration Vector
This lesson shows the geometric implication of the acceleration being in the plane defined by the unit tangent and unit normal vectors (at each instant). In this way, the user can see how the acceleration vector can be split into unit tangent and unit normal components. Specifically, the unit tangent component of acceleration measures changes in speed and the unit normal component of acceleration measures the change in direction (of the velocity). The four common examples are presented with animations that allow the user to see how this splitting looks at each point along each curve.
Subsection 7.8.1 Acceleration Components
In the interact below, the acceleration vector is orange, the unit tangent vector is red, and the unit normal vector is orange.
Exercises Exercises
Use the interactive tool above to help you answer the following questions.
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Consider the helix given by \(\mathbf{r}(t) = \langle 2\cos(t), 2\sin(t), t/3 \rangle \text{.}\) Change the value of \(t\) in the interact tool and observe the corresponding change in the unit tangnet, unit normal, and acceleration vectors. Based on your observations, answer the following:
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Is the speed of the object increasing, decreasing, or constant? Explain.
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Is the value of \(a_T\) (the magnitude of \(\mathbf{a}\) in the tangent direction) positive, negative, or 0? Explain.
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Is the value of \(a_N\) (the magnitude of \(\mathbf{a}\) in the normal direction) positive, negative, or 0? Explain.
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Describe the effect of the acceleration vector on the motion of this object.
