Question

After an object falls for \(t\) seconds, the speed \(S\) (in feet per second) of the object is recorded in the table. $$ \begin{array}{|l|c|c|c|c|c|c|c|} \hline t & 0 & 5 & 10 & 15 & 20 & 25 & 30 \\ \hline S & 0 & 48.2 & 53.5 & 55.2 & 55.9 & 56.2 & 56.3 \\ \hline \end{array} $$ (a) Create a line graph of the data. (b) Does there appear to be a limiting speed of the object? If there is a limiting speed, identify a possible cause.

Short answer

Yes, there appears to be a limiting speed. The possible cause is the air resistance acting against the object as it falls.

Step-by-step solution

Drawing the Line Graph

To create a line graph, each time-slice (t) will be plotted on the x-axis whereas the corresponding speed (S) will be plotted along the y-axis. Each point will represent a given time and speed pair. Now connect each point in chronological order, starting from the point at t=0.

Evaluating the graph for a limiting speed

On examining the line graph created, check for a limiting speed (a speed which the object cannot seem to exceed), meaning the speed at which the line begins to flatten and not increase by much.

Identifying a possible cause of the limiting speed

If there seems to be a limiting speed, a possible cause would be air resistance. As an object falls, the air resistance increases. Eventually, a point is reached where the air resistance equals the weight of the object. At this point, the object stops accelerating and continues falling at a constant speed - the limiting speed.

Key concepts

Line Graph of Speed Versus Time

Understanding the relationship between speed and time for moving objects is crucial for interpreting motion. A line graph is an effective visual tool for displaying this relationship. When plotting a line graph with time on the x-axis and speed on the y-axis, each point conveys the speed of an object at a particular moment. To draw the graph, you start by placing the time data along the horizontal axis and corresponding speed values vertically. After marking the points, such as (0, 0) for the object at rest and (5, 48.2) for the speed after 5 seconds, you connect them with lines to illustrate how the speed changes over time.

This visual representation allows for immediate identification of patterns, such as acceleration periods, constant speed, and even deceleration. It's imperative to pay attention to how the slope of the line changes. A steep slope indicates rapid acceleration, while a flatter slope suggests a slowing in acceleration. Through close examination of the slope, you can deduce how the object's velocity evolves over time, which is an essential aspect of interpreting motion graphs.

Evaluating Graphs for Limiting Behavior

Limiting behavior on a speed versus time graph reveals critical insights into an object's motion, particularly as it interacts with forces like air resistance. When evaluating the graph, if you notice the line leveling off as time passes, this horizontal asymptote suggests a limiting speed or terminal velocity. In such cases, the object is no longer accelerating, and the speed has become constant. To detect this trend, observe the intervals between plot points; when there is minimal to no change in speed over increasing time intervals, a limiting behavior is indicated.

It is beneficial to compare the increments in speed during the initial time periods with those later on. Early in the graph, if larger speed increases are represented by steeper slopes, and these slopes decrease over time until they almost flatten out, you're witnessing the object approaching its limiting speed. As an observer, your goal is to interpret such patterns to foresee the object's future motion and understand the forces at play.

Effect of Air Resistance on Falling Objects

Air resistance, an often-overlooked factor in motion, has a profound influence on the falling speed of objects. Air resistance opposes the gravitational pull acting on an object as it falls through the air, increasing with the object's speed until a critical point. When the force of air resistance equals the gravitational force, the object attains a state of dynamic equilibrium, and it no longer accelerates, falling at a constant speed known as terminal velocity.

This phenomenon is well-demonstrated through the limiting behavior observed in speed versus time graphs. As an object accelerates and its speed increases, the air resistance does too, until both forces balance each other out. At terminal velocity, the acceleration due to gravity is effectively neutralized by air resistance, and this balance is reflected as a flat, horizontal section on the graph where speed remains constant. Understanding this opposition between gravity and air resistance can help students grasp why objects do not continue to accelerate indefinitely as they fall.