Splitting the Acceleration Vector

This section contains materials that some instructors would skip in a traditional text, but we suggest spending a bit of time on this even if you typically steer away from physics applications. In particular, this section shows students the full extent of how to use their vector tools to efficiently calculate many useful quantities. While this is framed around understanding acceleration, this section helps to reinforce how measurements of a vectors direction and magnitude can be split and analyzed separately. This culminates with an efficient calculation of \(\vN\) and that wrap up our discussion of properties of the driver and properties of the road. offers a visual summary of the various vector quantities from this chapter.

Position is the central measurement of the study of motion, which is known as kinematics. Velocity and acceleration are the first and second derivatives of position, respectively, and are frequently important properties in the study of physics and mechanics. To better understand velocity as a vector-valued function of time, we examined how the magnitude and direction of position change separately. We separated velocity into its critical parts: speed (magnitude of velocity) and direction of travel (unit tangent vector). Speed is a scalar measurement for how fast the position of an object is changing as a function of time, while \(\vT\) is a unit vector that measures the direction in which position is changing.

As a vector quantity, acceleration is related to many important physical laws and descriptions. For example, Newton's Second Law of Motion relates the forces acting on an object to its acceleration through the formula \(\vec{F}_{net}= m \vec{a}\). This is often summarized as the sum of the forces acting on an object is the mass times the acceleration of the object. This description relates a combination of forces on a object to its position and velocity through integration of \(\va\). For an example of these calculations, refer back to .

To understand acceleration better as a vector quantity, we will examine acceleration as a change in the magnitude and direction of the velocity vector. In this section, we will use our knowledge of calculus and vector measurements to split the acceleration vector into parts related to the changes in the magnitude and direction of the velocity vector. We will also consider how these measurements relate to physical contexts.

The preview activity shows how you can graphically separate the acceleration vector into parts in the direction of travel and direction of turning. In this section, we will look at how to calculate these separate parts algebraically, understand what is being measured by these quantities, and show how these tools will give us important physical measurements as well as an efficient tool for calculating \(\vN\), the unit normal vector to a curve.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

Recall that by the definition of \(\vT\), \(\vv\) is parallel to \(\vT\) when \(\vv\) exists and is nonzero. Additionally, states that \(\vT \cdot \vN =0\). In other words, all of the velocity is in a direction parallel to the unit tangent vector and the unit normal vector is always orthogonal to the unit tangent vector to a curve, when they exist. In this subsection, we will look at splitting the acceleration vector into parts in the direction of travel and the direction of turning.

Suppose \(\va(t)\) is the acceleration vector for the motion along a curve in \(\mathbb{R}^3\). From , we have the TNB frame as a way of describing key directions related to motion along a curve in space. We would like to split the acceleration vector into a part in each of directions corresponding to \(\vT\), \(\vN\), and \(\vB\). Algebraically, we want to find scalars \(c_1\), \(c_2\), and \(c_3\) such that \(\va= c_1 \vT + c_2 \vN +c_3 \vB\). To do so, we will make use of several key facts from : \[\begin{aligned}\vT \cdot \vN \amp=0 \amp \vT \cdot \vB \amp =0 \amp \vN \cdot \vB \amp=0 \amp\vecmag{\vT}\amp=\vecmag{\vN}=\vecmag{\vB}=1\end{aligned}\]. If we compute the dot product of each of \(\vT\), \(\vN\), and \(\vB\) with the expression for \(\va\), we find \[\begin{aligned}\va \cdot \vT \amp= (c_1 \vT + c_2 \vN +c_3 \vB)\cdot \vT \\ \amp= c_1 (\vT \cdot \vT) + c_2 (\vN \cdot \vT) +c_3 (\vB \cdot \vT) \\ \amp=c_1 \\ \va \cdot \vN \amp= (c_1 \vT + c_2 \vN +c_3 \vB)\cdot \vN \\ \amp= c_1 (\vT \cdot \vN) + c_2 (\vN \cdot \vN) +c_3 (\vB \cdot \vN) \\ \amp=c_2 \\ \va \cdot \vB \amp= (c_1 \vT + c_2 \vN +c_3 \vB)\cdot \vB \\ \amp= c_1 (\vT \cdot \vB) + c_2 (\vN \cdot \vB) +c_3 (\vB \cdot \vB) \\ \amp=c_3\end{aligned}\] These computations demonstrate that because the TNB frame directions are all unit vectors that are orthogonal to each other, we can compute the amount of acceleration in each of these directions with a dot product!

Note that nothing in the argument for depended on being in three dimensions, and all of the calculus tools used in the argument work for any \(\mathbb{R}^n\).

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.

You saw in that calculating \(\vN\) for even a curve with a very nice parameterization algebraically using the definition requires extensive algebraic manipulation. Before going into an exampmle of the computations involved in splitting the acceleration vector, we will identify a relationship of the vectors involved in splitting the acceleration vector that yields another way of approaching \(\vN\). To do this, we consider the right triangle of vectors in . If we apply the Pythagorean Theorem to this right triangle, we can then use that the unit tangent vector and unit normal vector both have length one to conclude that \(\vecmag{ \va }^2 =(a_{\vT})^2+(a_{\vN})^2\).

Now we can solve for \(a_{\vN}\) and find \[\begin{aligned}\end{aligned}\]. Notice that since \(a_{\vN}\) is always nonnegative, we know that we can take the positive square root when solving here. We will soon use this formula to help find \(a_{\vN}\). Since \(\va = a_{\vT}\vT + a_{\vN}\vN\), we can also solve for \(\vN\) and have \[\begin{aligned}\end{aligned}\] which we will use to help find \(\vN\).

The method used in the previous example can be applied broadly and shows how to calculate all parts of in terms of \(\vv\) and \(\va\) (for both an interval or a single value of \(t\)).

\[\begin{aligned}\vT \amp= \frac{\vv}{\vecmag{\vv}} \\ a_{\vT} \amp= \frac{\va\cdot\vv}{\vecmag{\vv}} \Rightarrow a_{\vT} \vT = \frac{\va\cdot \vv}{\vv \cdot \vv} \vv \\ a_{\vN} \amp= \vecmag{ \va - a_{\vT} \vT } = \sqrt{\vecmag{ \va }^2 - (a_{\vT})^2}= \sqrt{(\va \cdot \va)- \frac{(\va\cdot\vv)^2}{\vecmag{\vv}}} \\ a_{\vN} \vN \amp= \va - a_{\vT} \vT \\ \vN \amp= \frac{\va - a_{\vT} \vT}{a_{\vN}}\end{aligned}\]

, , and offer conceptual and physical meaning to the different measurements involved in the splitting of acceleration. We will take a moment now to relate \(a_{\vN}\), the acceleration due to a change in the direction of travel, and \(\kappa\), the curvature, which measures rate of change for the direction of travel on a curve.

Condensed — the full section is in Boelkins et al., Active Calculus Multivariable.