5.10: Segmented kinematics
- Page ID
- 103454
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Sometimes motion occurs in different time segments or space segments. For example a car might accelerate, and at a certain time \(t_{1}\) it brakes and continues at a constant velocity (see Figure 5.12). This kind of motion is called segmented motion and can be described by segmented or piece-wise functions like this:
\[s_{A}(t)= \begin{cases}f_{1}(t) & t \leq t_{1} \tag{5.98} \label{5.98}\\[4pt] f_{2}(t) & t>t_{1}\end{cases}\]
The velocity and acceleration can be found by differentiation on the different segments as follows:
\[\begin{align} & v_{s, A}(t)=\dot{s}_{A}(t)= \begin{cases}\dot{f}_{1}(t) & t \leq t_{1} \\[4pt] \dot{f}_{2}(t) & t>t_{1}\end{cases} \tag{5.99} \label{5.99}\\[4pt] & a_{s, A}(t)=\ddot{s}_{A}(t)= \begin{cases}\ddot{f}_{1}(t) & t \leq t_{1} \\[4pt] \ddot{f}_{2}(t) & t>t_{1}\end{cases} \tag{5.100} \label{5.100}\end{align}\]
The procedure is similar if the motion is segmented along the path coordinate \(s\) instead of time \(t\), using functions \(f_{1}(s)\) and \(f_{2}(s)\).
When integrating the segmented motion, one proceeds by integrating of the first segment, and then taking the end position and velocity of the first segment as initial condition for the integration of the next segment. It is important to note that it is impossible for a point mass to instantaneously ’jump’ or ’teleport ’ in space. For that reason \(f_{1}\left(t_{1}\right)=f_{2}\left(t_{1}\right)\) needs to be obeyed. Moreover, from Newton’s second law and the inertia of mass, we know that it also takes time to change the velocity of an object. Therefore often also \(\dot{f}_{1}\left(t_{1}\right)=\dot{f}_{2}\left(t_{1}\right)\) holds for the velocities to be equal at the interface between the segments. There are exceptions in cases like collisions, where very high forces occur, such that the accelerations are so high that a substantial velocity change can occur suddenly.