WVWC-AEM Arc Length and Components of Acceleration

Section 3.3 Arc Length and Components of Acceleration

In this section we review the concepts of arc length and components of acceration, two important quantities in the description of motion in higher dimensions.

Subsection Arc Length

Given a smooth space curve over some interval \(I\) (see Definition 3.2.2), we can find the corresponding arc length over \(I\text{.}\)

Definition 3.3.1. Arc Length.

Let \(\rr(t)\) be a smooth space curve defined over the interval \(I\) and let \(a,b\in I\text{.}\) The arc length of \(\rr(t)\) from \(t = a\) to \(t = b\) is the integral

\begin{equation*} s = \int_a^b \norm{\rrp(t)}\dd{t}\text{.} \end{equation*}

Example 3.3.2. Arc Length of a Parabolic Segment.

Find the integral that gives the length of the parabolic segment \(y = x^2\) from \(x = 0\) to \(x = 1\text{.}\)

Solution.

Hint: set \(\rr(t) = \mqty[t & t^2]\) and use Definition 3.3.1.

Another application of the arc length integral is in the parameterization of curves with respect to arc length. In particular, if \(\rr(t)\) is smooth over \(I = [a, b]\) and \(t\in I\text{,}\) then we can define an arc length function \(s(t)\) by

\begin{equation*} s(t) = \int_a^b\norm{\rrp(t)}\dd{t}\text{.} \end{equation*}

This function is one-to-one and therefore invertible on \(I\) and so has an inverse function \(t(s)\text{.}\) The arc length parameterization of the space curve \(\rr(t)\) is then defined to be

\begin{equation*} \rr(t(s))\text{.} \end{equation*}

In terms of motion, the arc length parameterization corresponds to moving along the curve \(\rr\) at unit velocity at all times. This is because

\begin{equation*} \dv{\rr}{s} = \dv{\rr}{t}\dd{t}{s} = \TT(t)\text{.} \end{equation*}

Subsection Components of Acceleration

If a space curve \(\rr(t)\) is twice-differentiable then its acceleration is given by \(\vb{a}(t) = \rr''(t)\text{.}\) If the space curve is also smooth then we can find how much of the acceleration is parallel to the velocity.

Definition 3.3.3. Tangential Component of Acceleration.

Let \(\rr(t)\) denote a smooth, twice-differentiable space curve with corresponding acceleration vector \(\vb{a}(t) = \rr''(t)\) and unit tangent vector \(\TT(t)\text{.}\) The tangential component of acceleration is defined to be the vector \(\vb{a}_{\text{tan}}\) given by

\begin{equation*} \vb{a}_{\text{tan}} = (\vb{a}\cdot\TT)\TT\text{.} \end{equation*}

We can also define the normal component of acceleration to be the vector \(\vb{a}_{\text{norm}}\) given by

\begin{equation*} \vb{a}_{\text{norm}} = \vb{a} - \vb{a}_{\text{tan}}\text{.} \end{equation*}

This vector is orthogonal to \(\TT\) (and therefore \(\vb{a}_{\text{tan}}\)), and the normal and tangential components of acceleration always sum to \(\vb{a}\text{.}\) This is related to the unit normal vector \(\vb{N}(t) = \frac{\TT^\prime(t)}{\norm{\TT^\prime(t)}}\) by

\begin{equation*} \vb{a}_{\text{norm}} = (\vb{a}\cdot\vb{N})\vb{N}\text{.} \end{equation*}

Advanced Engineering Mathematics Lecture Notes: West Virginia Wesleyan College

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