## Section2.2Tracking Motion

Tracing a motion on a diagram is the most basic way of representing motion. If you have taken a road trip or train journey, you are already familiar with this idea. You do this by marking time at various points along your path. This would be your motion diagram. For instance, Figure 2.2.1 shows motion diagram for a road trip from Boston (USA) to San Framcisco (USA), where points on the map are mrked every 24 hours.

Often we mark position at regular time intervals so that we can easily understand the motion diagram immediately qualitatively, such as where an object is moving steadily, or where speeding up, or where slowing down. For instance, from motion diagram in Figure 2.2.2, it is immediately obvious that the car is slowing between A and B, moving at a steady pace between B and C, and speeding up betwen C and D.

If you were moving on a straight path, as in Figure 2.2.2, you could view the motion occuring on a Cartesian axis, say the $x$ axis. Your markings would then give $x$ coordinates at various times $t \text{.}$ We can fit these data points to obtain a position as a function of time, namely, $x(t) \text{.}$ The benefit of thinking in terms of function $x(t)$ is that you can do further analysis of motion. You will see that this function gives us analytic definition of velocity.

Another way to look at motion along $x$ axis is to plot $x$ versus $t\text{.}$ Figure 2.2.3 shows the $x$ vs $t$ plot corresponding to the motion of Figure 2.2.2. By connecting the points, or interpolating between the marked points, we obtain a continuous function $x(t)$ from the plot, as shown.

Figure 2.2.5 shows a boy running on a straight track. The position of the boy is marked every 2 seconds. The line below the sketch shows the scale of length in the drawing.

(a) Draw $x$ versus $t$ representation on the motion diagram.

(b) Is the motion steady, slowing down, or speeding up? How can you tell?

Hint

Use the scale to figure out distances from where the boy is at point marked 1.

(a) See solution. (b) Steady since covering same distance in same amount of time.

Solution 1 (a)

(a) The plot in Figure 2.2.6 shows $x$ values along the ordinate and $t$ values along the abscissa.

Solution 2 (b)

(b) Steady since covering same distance in same amount of time. Also, steady since the plot of $x$ vs $t$ is a straight line.

Figure 2.2.8 shows a plot of $x$ versus $t \text{.}$ Draw a motion diagram that will result in this plot. Show space scales and indicate value of time on your diagram.

Hint

Mark points on the ordinate at equal-time intervals.

First we mark points on the ordinate axis at equal-time intervals. We choose to mark points separated by $10\text{ s}$ intervals. These points are shown as dots in $x$ vs $t$ plot in Figure 2.2.9. Then we draw points on the $x$ axis only to get the motion diagram, where we now label each point as 1, 2, 3, ..., which stand for the instants, with label 2 for 10 sec, label 3 for 20 sec, and so forth.
Figure 2.2.11 shows a plot of $x$ versus $t \text{.}$ Draw a motion diagram that will result in this plot. Show space scales and indicate value of time on your diagram.