Motion Graphs for Zero Acceleration
Graph of Position vs. Time (a = 0, so v is constant)
Time is usually an independent variable that other quantities, such as position, depend upon. A graph of position versus time would, thus, have \(x\) on the vertical axis and \(t\) on the horizontal axis. The figure below is just such a straight-line graph. It shows a graph of position versus time for a jet-powered car on a very flat dry lake bed in Nevada.
Graph of position versus time for a jet-powered car on the Bonneville Salt Flats. Image credit: OpenStax, Chemistry
Using the relationship between dependent and independent variables, we see that the slope in the graph above is average velocity \(\bar{v}\) and the intercept is position at time zero—that is, \(x_0\). Substituting these symbols into \(y = mx + b\) gives
\(x = \bar{v} t + x_0\)
or
\(x = x_0 + \bar{v} t.\)
Thus a graph of position versus time gives a general relationship among displacement(change in position), velocity, and time, as well as giving detailed numerical information about a specific situation.
The Slope of x vs. t
The slope of the graph of position \(x\) vs. time \(t\) is velocity \(v\).
\(\text{slope} = \cfrac{\Delta x}{\Delta t} = v\)
Notice that this equation is the same as that derived algebraically from other motion equations in the previous lessons.
From the figure we can see that the car has a position of 25 m at 0.50 s and 2000 m at 6.40 s. Its position at other times can be read from the graph; furthermore, information about its velocity and acceleration can also be obtained from the graph.
Example on Determining Average Velocity from a Graph of Position versus Time: Jet Car
Find the average velocity of the car whose position is graphed in the figure above.
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
\(\text{slope} = \cfrac{\Delta x}{\Delta t} = \bar{v}.\)
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.
\(\bar{v} = \cfrac{\Delta x}{\Delta t} = \cfrac{2000\text{ m} \; - \; 525\text{ m}}{6.4\text{ s} \; - \; 0.50\text{ s}},\)
yielding
\(\bar{v} = 250\text{ m/s}.\)
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.
This lesson is part of:
One-Dimensional Kinematics