2.8.2: Motion graphs of acceleration, velocity, and position versus time.
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Interpret motion graphs of acceleration, velocity, and position versus time as well as the relationship between these graphs.
Earlier in this chapter we examined some example problems about a moving train. We are now going to further examine the motion graphs that acompanied those example problems. The relevant motion graphs are repeated here in Figure \(\PageIndex{1}\).
In the position versus time graph, you will notice that the position increases throughout the region graphed, but not at a constant rate. At the start of the graph the slope of the curve is increasing as time increases. Eventually this slope appears to be constant for awhile, And then towards the end of the graph, the slope seems to decrease. It can sometimes be a challenge to interpret a graph of changing slope so here are a few things to consider in that regard. If we look at the first two marks on the graph, it appears as if the position has changed barely at all. We can imagine if the curve continued with this slope that it would take a very long time to even go as far as 50 m. If we look at the second and third points, the position has increased quite a bit more over a similar period of time. This indicates that the slope of the line is increasing.
In the velocity versus time graph we can see how the trends in the position versus time graph are indicated. The portion at the start of the velocity versus time graph is increasing, which corresponds to the portion of the position versus time graph where the slope is changing. This makes sense because the velocity is the slope of the position versus time graph. If the slope is not constant, then the velocity should not be constant either. However, it does appear to be constantly increasing. In the middle section of the position versus time graph where the slope is constant we see that we have a constant velocity on the velocity versus time graph. This is consistent with what we saw on motion graphs earlier in this chapter. Finally, we see at the end of the velocity versus time graph a downward sloping straight line. This is consistent with what we had seen earlier for the position versus time graph with a decrease in the slope of the graph towards the end. Because the slope is decreasing, the velocity is also decreasing. Note that the velocity is not negative here, simply becoming smaller. (What would you expect to see if the velocity were negative?)
The final graph is acceleration versus time. In this graph we see only flat lines, but different values for those flat lines. At the start we see a positive flat line for the value of acceleration. This corresponds to the section of the velocity versus time graph where the velocity is increasing. In the middle section where the velocity is constant, we see a flat line at exactly zero acceleration. At the end of the graph where the velocity is decreasing, we see a negative flat line. If we consider that acceleration is a measure of the change in velocity with time, all of these results should be consistent. When the velocity is increasing, the acceleration is positive. When the velocity is constant, the acceleration is zero. When the velocity is decreasing, the acceleration is negative.
You might be wondering what would happen to all of these graphs if the acceleration were changing incrementally as we had seen for velocity and position in previous examples. That is possible, but beyond the scope of this class. We will only consider examples of constant acceleration or stepping between constant accelerations. Sometimes that acceleration will be positive. Sometimes it will be negative. And sometimes it will be zero. We will next consider how both position and velocity might change over a longer period of time when they have a constant acceleration
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Curated from resources found in Introduction to Physics published by OpenStax.