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College Physics 2e

2.8 Graphical Analysis of One-Dimensional Motion

College Physics 2e2.8 Graphical Analysis of One-Dimensional Motion

Learning Objectives

By the end of this section, you will be able to:

  • Describe a straight-line graph in terms of its slope and y-intercept.
  • Determine average velocity or instantaneous velocity from a graph of position vs. time.
  • Determine average or instantaneous acceleration from a graph of velocity vs. time.
  • Derive a graph of velocity vs. time from a graph of position vs. time.
  • Derive a graph of acceleration vs. time from a graph of velocity vs. time.

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 xx-axis and the vertical axis the yy-axis, as in Figure 2.44, a straight-line graph has the general form

y=mx+b.y=mx+b.
2.89

Here mm is the slope, defined to be the rise divided by the run (as seen in the figure) of the straight line. The letter bb is used for the y-intercept, which is the point at which the line crosses the vertical axis.

Graph of a straight-line sloping up at about 40 degrees.
Figure 2.44 A straight-line graph. The equation for a straight line is y = mx + b y = mx + b .

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 xx on the vertical axis and tt on the horizontal axis. Figure 2.45 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.

Line graph of jet car displacement in meters versus time in seconds. The line is straight with a positive slope. The y intercept is four hundred meters. The total change in time is eight point zero seconds. The initial position is four hundred meters. The final position is two thousand meters.
Figure 2.45 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 v-v- and the intercept is position at time zero—that is, x0x0. Substituting these symbols into y=mx+by=mx+b gives

x = v - t + x 0 x = v - t + x 0
2.90

or

x=x0+v-t.x=x0+v-t.
2.91

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 xx vs. time tt is velocity vv.

slope = Δ x Δt = v slope = Δ x Δt = v
2.92

Notice that this equation is the same as that derived algebraically from other motion equations in Motion Equations for Constant Acceleration in One Dimension.

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 2.17

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 2.45.

Strategy

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

slope = Δ x Δt = v- . slope = Δ x Δt = v- .
2.93

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 xx and tt values of the chosen points into the equation. Remember in calculating change (Δ)(Δ) we always use final value minus initial value.

v-= Δx Δt =2000 m525 m6.4 s0.50 s,v-= Δx Δt =2000 m525 m6.4 s0.50 s,
2.94

yielding

v-=250 m/s.v-=250 m/s.
2.95

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.

Graphs of Motion when aa is constant but a0a0

The graphs in Figure 2.46 below represent the motion of the jet-powered car as it accelerates toward its top speed, but only during the time when its acceleration is constant. Time starts at zero for this motion (as if measured with a stopwatch), and the position and velocity are initially 200 m and 15 m/s, respectively.

Three line graphs. First is a line graph of displacement over time. Line has a positive slope that increases with time. Second line graph is of velocity over time. Line is straight with a positive slope. Third line graph is of acceleration over time. Line is straight and horizontal, indicating constant acceleration.
Figure 2.46 Graphs of motion of a jet-powered car during the time span when its acceleration is constant. (a) The slope of an xx vs. tt 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 vv vs. tt graph is constant for this part of the motion, indicating constant acceleration. (c) Acceleration has the constant value of 5.0 m/s25.0 m/s2 over the time interval plotted.
A photo shows a U S Air Force jet speeding down a track with a lot of smoke around it.
Figure 2.47 A U.S. Air Force jet car speeds down a track. (credit: Matt Trostle, Flickr)

The graph of position versus time in Figure 2.46(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 2.46(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 2.46(b) is obtained. Furthermore, the slope of the graph of velocity versus time is acceleration, which is shown in Figure 2.46(c).

Example 2.18

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 xx vs. tt graph in the graph below.

A graph of displacement versus time for a jet car. The x axis for time runs from zero to thirty five seconds. The y axis for displacement runs from zero to three thousand meters. The curve depicting displacement is concave up. The slope of the curve increases over time. Slope equals velocity v. There are two points on the curve, labeled, P and Q. P is located at time equals ten seconds. Q is located and time equals twenty-five seconds. A line tangent to P at ten seconds is drawn and has a slope delta x sub P over delta t sub p. A line tangent to Q at twenty five seconds is drawn and has a slope equal to delta x sub q over delta t sub q. Select coordinates are given in a table and consist of the following: time zero seconds displacement two hundred meters; time five seconds displacement three hundred thirty eight meters; time ten seconds displacement six hundred meters; time fifteen seconds displacement nine hundred eighty eight meters. Time twenty seconds displacement one thousand five hundred meters; time twenty five seconds displacement two thousand one hundred thirty eight meters; time thirty seconds displacement two thousand nine hundred meters.
Figure 2.48 The slope of an xx vs. tt 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 2.48, where Q is the point at t=25 st=25 s.

Solution

1. Find the tangent line to the curve at t=25 st=25 s.

2. 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.

3. Plug these endpoints into the equation to solve for the slope, vv.

slope = v Q = Δx Q Δt Q = 3120 m 1300 m 32 s 19 s slope = v Q = Δx Q Δt Q = 3120 m 1300 m 32 s 19 s
2.96

Thus,

v Q = 1820 m 13 s = 140 m/s. v Q = 1820 m 13 s = 140 m/s.
2.97

Discussion

This is the value given in this figure’s table for vv at t=25 st=25 s. The value of 140 m/s for vQvQ is plotted in Figure 2.48. The entire graph of vv vs. tt can be obtained in this fashion.

Carrying this one step further, we note that the slope of a velocity versus time graph is acceleration. Slope is rise divided by run; on a vv vs. tt graph, rise = change in velocity ΔvΔv and run = change in time ΔtΔt.

The Slope of v vs. t

The slope of a graph of velocity vv vs. time tt is acceleration aa.

slope = Δv Δt = a slope = Δv Δt = a
2.98

Since the velocity versus time graph in Figure 2.46(b) is a straight line, its slope is the same everywhere, implying that acceleration is constant. Acceleration versus time is graphed in Figure 2.46(c).

Additional general information can be obtained from Figure 2.48 and the expression for a straight line, y=mx+by=mx+b.

In this case, the vertical axis yy is VV, the intercept bb is v0v0, the slope mm is aa, and the horizontal axis xx is tt. Substituting these symbols yields

v=v0+at.v=v0+at.
2.99

A general relationship for velocity, acceleration, and time has again been obtained from a graph. Notice that this equation was also derived algebraically from other motion equations in Motion Equations for Constant Acceleration in One Dimension.

It is not accidental that the same equations are obtained by graphical analysis as by algebraic techniques. In fact, an important way to discover physical relationships is to measure various physical quantities and then make graphs of one quantity against another to see if they are correlated in any way. Correlations imply physical relationships and might be shown by smooth graphs such as those above. From such graphs, mathematical relationships can sometimes be postulated. Further experiments are then performed to determine the validity of the hypothesized relationships.

Graphs of Motion Where Acceleration is Not Constant

Now consider the motion of the jet car as it goes from 165 m/s to its top velocity of 250 m/s, graphed in Figure 2.49. Time again starts at zero, and the initial velocity is 165 m/s. (This was the final velocity of the car in the motion graphed in Figure 2.46.) Acceleration gradually decreases from 5.0 m/s25.0 m/s2 to zero when the car hits 250 m/s. The velocity increases until 55 s and then becomes constant, since acceleration decreases to zero at 55 s and remains zero afterward.

Two line graphs of jet car velocity and acceleration, respectively. First line graph is of velocity over time. Line graph has a positive slope that decreases over time and flattens out at the end. Second line graph is of acceleration over time. Line has a negative slope that increases over time until it flattens out at the end. The line is not smooth, but has several kinks.
Figure 2.49 Graphs of motion of a jet-powered car as it reaches its top velocity. This motion begins where the motion in Figure 2.46 ends. (a) The velocity gradually approaches its top value. The slope of this graph is acceleration. It is plotted in the final graph. (b) Acceleration gradually declines to zero when velocity becomes constant. Notice in each of the three graphs that the acceleration drops down to zero and the velocity levels out. This results in a position-time graph that is almost linear. A close-up of the position time graph would show a slight curvature, as indicated in the velocity graph.

Example 2.19

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 vv vs. tt graph in Figure 2.49(a).

Strategy

The slope of the curve at t=25 st=25 s is equal to the slope of the line tangent at that point, as illustrated in Figure 2.49(a).

Solution

Determine endpoints of the tangent line from the figure, and then plug them into the equation to solve for slope, aa.

slope = Δv Δt = 260 m/s 210 m/s 51 s 1.0 s slope = Δv Δt = 260 m/s 210 m/s 51 s 1.0 s
2.100
a=50 m/s50 s=1.0 m/s2.a=50 m/s50 s=1.0 m/s2.
2.101

Discussion

Note that this value for aa is consistent with the value plotted in Figure 2.49(b) at t=25 st=25 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?

Line graph of velocity versus time. The line has three legs. The first leg is flat. The second leg has a negative slope. The third leg also has a negative slope, but the slope is not as negative as the second leg.
Figure 2.50
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