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where we take the positive value as the physically relevant answer. Thus, it takes about 2.5 seconds for the piece of ice to hit the water.
PHET EXPLORATIONS
Equation Grapher
Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. ) to see how they add to generate the polynomial curve.
Click to view content (https://phet.colorado.edu/sims/equation-grapher/equation-grapher_en.html) Figure 2.
2.8 Graphical Analysis of One-Dimensional Motion
A graph, like a picture, is worth a thousand words. Graphs not only contain numerical information; they also reveal relationships between physical quantities. This section uses graphs of position, velocity, and acceleration versus time to illustrate one-dimensional kinematics.
Slopes and General Relationships
First note that graphs in this text have perpendicular axes, one horizontal and the other vertical. When two physical quantities are plotted against one another in such a graph, the horizontal axis is usually considered to be an independent variable and the vertical axis a dependent variable. If we call the horizontal axis the -axis and the vertical axis the -axis, as in Figure 2.46, a straight-line graph has the general form
Here is the slope , defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter is used for the y-intercept , which is the point at which the line crosses the vertical axis.
Figure 2.46 A straight-line graph. The equation for a straight line is.
Graph of Position vs. Time (a = 0, sov 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 on the vertical axis and on the horizontal axis. Figure 2.47 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.
2.8 • Graphical Analysis of One-Dimensional Motion 77
Figure 2.47 Graph of position versus time for a jet-powered car on the Bonneville Salt Flats.
Using the relationship between dependent and independent variables, we see that the slope in the graph above is average velocity and the intercept is position at time zero—that is,. Substituting these symbols into gives
or
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.
From the figure we can see that the car has a position of 525 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 2.
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 vs. is average velocity, since slope equals rise over run. In this case, rise = change in position and run = change in time, so that
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.)
The Slope ofx vs.t
The slope of the graph of position vs. time is velocity.
Notice that this equation is the same as that derived algebraically from other motion equations in Motion Equations for Constant Acceleration in One Dimension.
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Figure 2.48 Graphs of motion of a jet-powered car during the time span when its acceleration is constant. (a) The slope of an vs. 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 vs. graph is constant for this part of the motion, indicating constant acceleration. (c) Acceleration has the constant value of over the time interval plotted.
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Figure 2.49 A U.S. Air Force jet car speeds down a track. (credit: Matt Trostle, Flickr)
The graph of position versus time in Figure 2.48(a) is a curve rather than a straight line. The slope of the curve becomes steeper as time progresses, showing that the velocity is increasing over time. The slope at any point on a position-versus-time graph is the instantaneous velocity at that point. It is found by drawing a straight line tangent to the curve at the point of interest and taking the slope of this straight line. Tangent lines are shown for two points in Figure 2.48(a). If this is done at every point on the curve and the values are plotted against time, then the graph of velocity versus time shown in Figure 2.48(b) is obtained. Furthermore, the slope of the graph of velocity versus time is acceleration, which is shown in Figure 2.48(c).
EXAMPLE 2.
Determining Instantaneous Velocity from the Slope at a Point: Jet Car
Calculate the velocity of the jet car at a time of 25 s by finding the slope of the vs. graph in the graph below.
Figure 2.50 The slope of an vs. graph is velocity. This is shown at two points. Instantaneous velocity at any point is the slope of the tangent at that point.
Strategy
The slope of a curve at a point is equal to the slope of a straight line tangent to the curve at that point. This principle is illustrated in Figure 2.50, where Q is the point at.
Solution
- Find the tangent line to the curve at.
- Determine the endpoints of the tangent. These correspond to a position of 1300 m at time 19 s and a position of 3120 m at time 32 s.
- Plug these endpoints into the equation to solve for the slope,.
2.8 • Graphical Analysis of One-Dimensional Motion 81
Figure 2.51 Graphs of motion of a jet-powered car as it reaches its top velocity. This motion begins where the motion in Figure 2.48 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 2.
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 vs. graph in Figure 2.51(b).
Strategy
The slope of the curve at is equal to the slope of the line tangent at that point, as illustrated in Figure 2.51(b).
2.8 • Graphical Analysis of One-Dimensional Motion 83
Solution Determine endpoints of the tangent line from the figure, and then plug them into the equation to solve for slope,.
Discussion Note that this value for is consistent with the value plotted in Figure 2.51(c) at.
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?
Figure 2.
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.
Figure 2.
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