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Graphical Analysis of One-Dimensional Motion

Graphs of Motion When a is Constant but a≠0

The graphs in Figure 2.48 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 2.48 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 \(5.0 \, m/s^{2}\) 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 2.49 A U.S. Air Force jet car speeds down a track


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 2.18 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.

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 2.50 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 2.50, where \(Q\) is the point at \(t=25 \, s\).

Solution

1. Find the tangent line to the curve at \(t=25 \, s\).

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, \(v\).

\(\text { slope }=v_{\mathrm{Q}}=\frac{\Delta x_{\mathrm{Q}}}{\Delta t_{\mathrm{Q}}}=\frac{(3120 \mathrm{~m}-1300 \mathrm{~m})}{(32 \mathrm{~s}-19 \mathrm{~s})}\)

Thus,

\(v_{\mathrm{Q}}=\frac{1820 \mathrm{~m}}{13 \mathrm{~s}}=140 \mathrm{~m} / \mathrm{s}.\)

Discussion

This is the value given in this figure's table for \(v\) at \(t=25 \mathrm{~s}\). The value of \(140 \mathrm{~m} / \mathrm{s}\) for \(v_{\mathrm{Q}}\) is plotted in Figure 2.50. 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 \(\Delta v\) and run \(=\) change in time \(\Delta t\).

The Slope of V vs. \(T\)

The slope of a graph of velocity \(v\) vs. time \(t\) is acceleration \(a\).

\(\text { slope }=\frac{\Delta v}{\Delta t}=a\)

Since the velocity versus time graph in Figure 2.48(b) is a straight line, its slope is the same everywhere, implying that acceleration is constant. Acceleration versus time is graphed in Figure 2.48(c).

Additional general information can be obtained from Figure \(2.50\) and the expression for a straight line, \(y=m x+b\)

In this case, the vertical axis \(y\) is \(V\), the intercept \(b\) is \(v_{0}\), the slope \(m\) is \(a\), and the horizontal axis \(x\) is \(t\). Substituting these symbols yields

\(v=v_{0}+a t\)

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.