Motion Graphs for Non-Constant Acceleration
Graphs of Motion Where Acceleration is Not Constant
Now consider the motion of the jet car from the previous lesson (see image below) as it goes from 165 m/s to its top velocity of 250 m/s, graphed in the figure below.
A U.S. Air Force jet car speeds down a track. Image credit: Matt Trostle, Flick
Graphs of motion of a jet-powered car during the time span when its acceleration is constant. (a) The slope of an \(x\) vs. \(t\) graph is velocity. This is shown at two points, and the instantaneous velocities obtained are plotted in the next graph. Instantaneous velocity at any point is the slope of the tangent at that point. (b) The slope of the \(v\) vs. \(t\) graph is constant for this part of the motion, indicating constant acceleration. (c) Acceleration has the constant value of \(5.0\text{ m/s}^2\) over the time interval plotted. Image credit: OpenStax, College Physics
Time again starts at zero, and the initial position and velocity are 2900 m and 165 m/s, respectively. (These were the final position and velocity of the car in the motion graphed in the figure above from the previous lesson.) Acceleration gradually decreases from 5.0 m/s2 to zero when the car hits 250 m/s. The slope of the \(x\) vs. \(t\) graph increases until \(t = 55\text{ s},\) after which time the slope is constant. Similarly, velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward.
Graphs of motion of a jet-powered car as it reaches its top velocity. This motion begins where the motion in the first graph above ends. (a) The slope of this graph is velocity; it is plotted in the next graph. (b) The velocity gradually approaches its top value. The slope of this graph is acceleration; it is plotted in the final graph. (c) Acceleration gradually declines to zero when velocity becomes constant.
Example: Calculating Acceleration from a Graph of Velocity versus Time
Calculate the acceleration of the jet car at a time of 25 s by finding the slope of the \(v\) vs. \(t\) graph in figure (b) above.
Strategy
The slope of the curve at \(t = 25\text{ s}\) is equal to the slope of the line tangent at that point, as illustrated in figure (b) above.
Solution
Determine endpoints of the tangent line from the figure, and then plug them into the equation to solve for slope, \(a\).
\(\text{slope} = \cfrac{\Delta v}{\Delta t} = \cfrac{260\text{ m/s} \; - \; 210\text{ m/s}}{51\text{ s} \; - \; 1.0\text{ s}}\)
\(a = \cfrac{50\text{ m/s}}{50\text{ s}} = 1.0\text{ m/s}^2.\)
Discussion
Note that this value for \(a\) is consistent with the value plotted in figure (c) at \(t = 25\text{ s}\).
A graph of position versus time can be used to generate a graph of velocity versus time, and a graph of velocity versus time can be used to generate a graph of acceleration versus time. We do this by finding the slope of the graphs at every point. If the graph is linear (i.e., a line with a constant slope), it is easy to find the slope at any point and you have the slope for every point.
Graphical analysis of motion can be used to describe both specific and general characteristics of kinematics. Graphs can also be used for other topics in physics. An important aspect of exploring physical relationships is to graph them and look for underlying relationships.
Check Your Understanding
A graph of velocity vs. time of a ship coming into a harbor is shown below. (a) Describe the motion of the ship based on the graph. (b)What would a graph of the ship’s acceleration look like?
Image credit: OpenStax, College Physics
Solution
(a) The ship moves at constant velocity and then begins to decelerate at a constant rate. At some point, its deceleration rate decreases. It maintains this lower deceleration rate until it stops moving.
(b) A graph of acceleration vs. time would show zero acceleration in the first leg, large and constant negative acceleration in the second leg, and constant negative acceleration.
Image credit: OpenStax, College Physics
This lesson is part of:
One-Dimensional Kinematics