2.6.1 Thinking Graphically

A good way to solidify your intuitive understanding of motion is to consider the problem of a hard, smooth ball rolling on a smooth track. The track is made up of several straight segments connected together. Each segment may be either horizontal or inclined. Your task is to analyze the ball’s motion graphically.

There are a small number of rules to follow:

Assume that the ball passes smoothly from one segment of the track to the next, with no abrupt change of speed and without ever leaving the track.
The graphs have no numbers, but they should show the correct relationships. For example, the position graph should be steeper in regions of higher speed.
The position $s$ is the position measured along the track. Similarly, $v_{s}$ and $a_{s}$ are the velocity and acceleration parallel to the track.

EXAMPLE 2.15 From track to graphs

Draw position, velocity, and acceleration graphs for the ball on the smooth track of FIGURE 2.32.

The figure illustrates the motion of a ball along a track made of different segments.

FIGURE 2.32 Full Alternative Text

VISUALIZE It is often easiest to begin with the velocity. There is no acceleration on the horizontal surface $(a_{s} = 0 if θ = 0 °)$ , so the velocity remains constant at $v_{0 s}$ until the ball reaches the slope. The slope is an inclined plane where the ball has constant acceleration. The velocity increases linearly with time during constant-acceleration motion. The ball returns to constant-velocity motion after reaching the bottom horizontal segment. The middle graph of FIGURE 2.33 shows the velocity.

The figure illustrates three graphs to depict the position, velocity, and acceleration of a ball moving on a smooth track.

FIGURE 2.33 Full Alternative Text

We can easily draw the acceleration graph. The acceleration is zero while the ball is on the horizontal segments and has a constant positive value on the slope. These accelerations are consistent with the slope of the velocity graph: zero slope, then positive slope, then a return to zero. The acceleration cannot really change instantly from zero to a nonzero value, but the change can be so quick that we do not see it on the time scale of the graph. That is what the vertical dashed lines imply.

Finally, we need to find the position-versus-time graph. The position increases linearly with time during the first segment at constant velocity. It also does so during the third segment of motion, but with a steeper slope to indicate a faster velocity. In between, while the acceleration is nonzero but constant, the position graph has a parabolic shape. Notice that the parabolic section blends smoothly into the straight lines on either side. An abrupt change of slope (a “kink”) would indicate an abrupt change in velocity and would violate rule 1.

EXAMPLE 2.16 From graphs to track

FIGURE 2.34 shows a set of motion graphs for a ball moving on a track. Draw a picture of the track and describe the ball’s initial condition. Each segment of the track is straight, but the segments may be tilted.

The figure illustrates three graphs to depict the position, velocity, and acceleration of a ball moving on a track.

FIGURE 2.34 Full Alternative Text

VISUALIZE The ball starts with initial velocity $v_{0 s} > 0$ and maintains this velocity for awhile; there’s no acceleration. Thus the ball must start out rolling to the right on a horizontal track. At the end of the motion, the ball is again rolling on a horizontal track (no acceleration, constant velocity), but it’s rolling to the left because $v_{s}$ is negative. Further, the final speed $(| v_{s} |)$ is greater than the initial speed. The middle section of the graph shows us what happens. The ball starts slowing with constant acceleration (rolling uphill), reaches a turning point (s is maximum, $v_{s} = 0)$ , then speeds up in the opposite direction (rolling downhill). This is still a negative acceleration because the ball is speeding up in the negative s-direction. It must roll farther downhill than it had rolled uphill before reaching a horizontal section of track. FIGURE 2.35 shows the track and the initial conditions that are responsible for the graphs of Figure 2.34.

FIGURE 2.35 Full Alternative Text

2.6.1 Thinking Graphically

EXAMPLE 2.15 From track to graphs

FIGURE 2.32 A ball rolling along a track.

FIGURE 2.33 Motion graphs for the ball in Example 2.15.

EXAMPLE 2.16 From graphs to track

FIGURE 2.34 Motion graphs of a ball rolling on a track of unknown shape.

FIGURE 2.35 Track responsible for the motion graphs of Figure 2.34.