Speed–time graphs

Journeys and average speed

<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\text{Distance}=\int_{t_1}^{t_2}|v|\,\mathrm dt\qquad\text{Average speed}=\dfrac{\text{distance}}{t_2-t_1}\quad(t_2&gt;t_1)\)</p></div></div>
<p>Find direction changes from v = 0 and a sign check. Evaluate displacement at the endpoints and each direction change, then add the absolute changes.</p>
<p class="paragraph-header">Motion diagram</p>
<div class="graph-block" data-graph-config="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" data-width="440" data-height="110" data-proportional style="width:440px;height:110px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:440px;--graph-aspect:440 / 110;"></div>
<p>\(\text{Change in displacement}=-3-0=-3\qquad\text{Distance}=5+8=13\)</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 40.7407%;"><col style="width: 59.2593%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Quantity</strong></th><th style="text-align: center; --cell-justify: center"><strong>How to find it</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Change in displacement from \(0\) to \(n\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Distance travelled in the \(n\)th second</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Total distance in the first \(n\) seconds</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Average speed</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Instantaneous speed</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>For the nth second use n − 1 to n, splitting again if direction changes. A rest time with no direction change needs no split.</p>

\(\text{Distance}=\int_{t_1}^{t_2}|v|\,\mathrm dt\qquad\text{Average speed}=\dfrac{\text{distance}}{t_2-t_1}\quad(t_2>t_1)\)

Find direction changes from v = 0 and a sign check. Evaluate displacement at the endpoints and each direction change, then add the absolute changes.

\(\text{Change in displacement}=-3-0=-3\qquad\text{Distance}=5+8=13\)

Change in displacement from \(0\) to \(n\)

\(s(n)-s(0)\) or \(\displaystyle\int_0^n v\,\mathrm{d}t\)

\(\left|s(n)-s(n-1)\right|\) (split further if the particle rests inside that second)

Total distance in the first \(n\) seconds

Split at every rest time; add \(\left|\text{change in }s\right|\) for each stage

\(\dfrac{\text{total distance}}{\text{total time}}\)

For the nth second use n − 1 to n, splitting again if direction changes. A rest time with no direction change needs no split.

Total Distance, Average Speed and Motion Diagrams

Total Distance, Average Speed and Motion Diagrams

Quantity	How to find it
Change in displacement from \(0\) to \(n\)
Distance travelled in the \(n\)th second
Total distance in the first \(n\) seconds
Average speed
Instantaneous speed