Graphical Analysis of One-Dimensional Motion

Example 2.17 Determining Average Velocity from a Graph of Position versus Time: Jet Car

Find the average velocity of the car whose position is graphed in Figure 2.47.

Strategy

The slope of a graph of $x$ vs. $t$ is average velocity, since slope equals rise over run. In this case, rise = change in position and run = change in time, so that

$\text { slope }=\frac{\Delta x}{\Delta t}=\bar{v}.$

Since the slope is constant here, any two points on the graph can be used to find the slope. (Generally speaking, it is most accurate to use two widely separated points on the straight line. This is because any error in reading data from the graph is proportionally smaller if the interval is larger).

Solution

1. Choose two points on the line. In this case, we choose the points labeled on the graph: (6.4 s, 2000 m) and (0.50 s, 525 m). (Note, however, that you could choose any two points).

2. Substitute the $x$ and $t$ values of the chosen points into the equation. Remember in calculating change $(\Delta)$ we always use final value minus initial value.

$\bar{v}=\frac{\Delta x}{\Delta t}=\frac{2000 \mathrm{~m}-525 \mathrm{~m}}{6.4 \mathrm{~s}-0.50 \mathrm{~s}},$

yielding

$\bar{v}=250 \mathrm{~m} / \mathrm{s}.$

Discussion

This is an impressively large land speed (900 km/h, or about 560 mi/h): much greater than the typical highway speed limit of 60 mi/h (27 m/s or 96 km/h), but considerably shy of the record of 343 m/s (1234 km/h or 766 mi/h) set in 1997.