## Graphical Analysis of One-Dimensional Motion

Read this text for an introduction on graphical position, velocity, and acceleration with regards to one another. As you read, pay attention to Figure 2.46 which is an example of a linear graph is the graph of position versus time when acceleration is zero.

See an example of this type of graph in Figure 2.47. In this graph, we can determine the slope by picking two different points on the line, taking the change in y-value, and dividing it by the change in x-value between those two points. In this case, the unit for slope is m/s, which is the unit for velocity. Therefore, the slope for a graph of position versus time with zero acceleration is the average velocity of that object. See how to calculate the average velocity of an object from this type of graph in Example 2.17.

When acceleration is a non-zero constant, the graph of position versus time is no longer linear. You can see an example of this type of graph in Figure 2.48. Note that while the position versus time graph is not linear, the velocity versus time graph is linear. In the position versus time graph, the slope at any given point is the instantaneous velocity of the object. The instantaneous slope can be determined by drawing a tangent line at the desired point along the graph and determining slope. Pay attention to the tangent lines drawn in Figure 2.48 (a).

To determine instantaneous velocity at a given time when acceleration is a non-zero constant, take a look at Example 2.18. We can determine instantaneous velocities at multiple points along a position-time graph with constant non-zero acceleration and make a table relating these instantaneous velocities to the specified time along the x-axis where we found them. Then, we can use that table to plot velocity versus time. This process is demonstrated in Figure 2.48 (a) and (b). The slope of this linear graph has units $m/s^{2}$, which are acceleration units. Therefore, the slope of the velocity versus time graph is acceleration.

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?

Figure 2.52

#### 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.

A line graph of acceleration versus time. There are three legs of the graph. All three legs are flat and straight. The first leg shows constant acceleration of 0. The second leg shows a constant negative acceleration. The third leg shows a constant negative acceleration that is not as negative as the second leg.

Figure 2.53