\(\def \u#1{\,\mathrm{#1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\)

Graphing Velocity

It can also be helpful to graph the velocity of an object over time, in addition to the position. The two graphs are not independent: the velocity at any time is the slope of the position graph, as in the example on the right. A turning point on a position graph corresponds to the place on the velocity axis where the curve crosses the horizontal axis.

Because acceleration is defined as $$a={\Delta v\over \Delta t},$$

Consider this example to the right. The object has a positive acceleration until $t=4$, and then a much larger negative acceleration before it comes to a stop. Remember that positive acceleration is not the same as speeding up: sign is about direction. In the first two seconds, this object has a positive acceleration but a negative velocity: the signs are opposite, which means that the object is slowing down (moving closer to zero). From $t=2$ until $t=4$, the acceleration is still positive but now the velocity is also positive, so the object is speeding up (moving away from zero).