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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 each other 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 $x$-axis and the vertical axis the $y$-axis, as in Figure 2.58, a straight-line graph has the general form

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

Figure 2.58 A straight-line graph. The equation for a straight line is $y=\text{mx}+b\text{.}$

Graph of Displacement vs. Time (a = 0, so v is constant)

Time is usually an independent variable that other quantities, such as displacement, depend upon. A graph of displacement versus time would, thus, have $x$ on the vertical axis and $t$ on the horizontal axis. Figure 2.59 is just such a straight-line graph. It shows a graph of displacement versus time for a jet-powered car on a very flat dry lake bed in Nevada.

Figure 2.59 Graph of displacement 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 $\stackrel{-}{v}$ and the intercept is displacement at time zero—that is, $x_0\text{.}$ Substituting these symbols into $y=\text{mx}+b$ gives

or

Thus a graph of displacement versus time gives a general relationship among displacement, velocity, and time, as well as giving detailed numerical information about a specific situation.

From the figure, we can see that the car is initially at 400 m with a displacement of 0 m at 0.00 s , the car is at 525 m with a total displacement of 125 m at t = 0.50 s. Other values can be read from the graph.

Graphs of Motion when $a$ is constant but $a\ne 0$

The graphs in Figure 2.60 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 displacement and velocity are initially 200 m and 15 m/s, respectively.

Figure 2.60 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 $5\text{.}{\text{0 m/s}}^2$ over the time interval plotted.

Figure 2.61 A U.S. Air Force jet car speeds down a track. (Credit: Matt Trostle, Flickr)

The graph of displacement versus time in Figure 2.60(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 displacement-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 2.60(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 2.60(b) is obtained. Furthermore, the slope of the graph of velocity versus time is acceleration, which is shown in Figure 2.60(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 $x$ vs. $t$ graph in the graph below.

Figure 2.62 The slope of an $x$ vs. $t$ 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 2.62, where Q is the point at $t=\text{25 s.}$

Solution

1. Find the tangent line to the curve at $t=\text{25 s.}$.

2. Determine the endpoints of the tangent. These correspond to a position of 1,300 m at time 19 s and a position of 3,120 m at time 32 s.

3. Plug these endpoints into the equation to solve for the slope, $v\text{.}$

2.96 $$\text{slope}=v_{\mathrm{Q}}=\frac{{\mathrm{\Delta }x}_{\mathrm{Q}}}{{\mathrm{\Delta }t}_{\mathrm{Q}}}=\frac{\left(\text{3,120 m}-\text{1,300 m}\right)}{\left(\text{32 s}-\text{19 s}\right)}$$

Thus,

2.97 $$v_{\mathrm{Q}}=\frac{\text{1,820 m}}{\text{13 s}}=\text{140 m/s.}$$

Discussion

This is the value given in this figure's table for $v$ at $t=\text{25 s.}$ The value of 140 m/s for $v_{\mathrm{Q}}$ is plotted in Figure 2.62. The entire graph of $v$ vs. $t$ 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 $v$ vs. $t$ graph, rise = change in velocity $\mathrm{\Delta }v$ and run = change in time $\mathrm{\Delta }t\text{.}$

Since the velocity versus time graph in Figure 2.60(b) is a straight line, its slope is the same everywhere, implying that acceleration is constant. Acceleration versus time is graphed in Figure 2.60(c).

Additional general information can be obtained from Figure 2.62 and the expression for a straight line, $y=\text{mx}+b\text{.}$

In this case, the vertical axis $y$ is $V\text{,}$ the intercept $b$ is $v_0\text{,}$ the slope $m$ is $a\text{,}$ and the horizontal axis $x$ is $t\text{.}$ Substituting these symbols yields

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 2.63. Time again starts at zero, and the initial displacement and velocity are 2900 m and 165 m/s, respectively. These were the final displacement and velocity of the car in the motion graphed in Figure 2.60. Acceleration gradually decreases from $5\text{.}{\text{0 m/s}}^2$ to zero when the car hits 250 m/s. The slope of the $x$ vs. $t$ graph increases until $t=\text{55 s,}$ 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.

Figure 2.63 Graphs of motion of a jet-powered car as it reaches its top velocity. This motion begins where the motion in Figure 2.60 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.

A graph of displacement 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, that is, 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 2.64

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 2.65