Problem 117 - Calculus of a Single Variable

Question

Modeling Data The table lists the speeds \(S\) (in feet per second) of a falling object at various times \(t\) (in seconds). $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {0} & {5} & {10} & {15} & {20} & {25} & {30} \\ \hline S & {0} & {48.2} & {53.5} & {55.2} & {55.9} & {56.2} & {56.3} \\ \hline\end{array} $$ (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.

Step-by-Step Solution

Answer

From the graphical representation, it can be seen that the object tends to have a limiting speed of approximately 56.3 feet per second. The cause of this limiting speed may be due to air resistance, which counterbalances the force of gravity at high speeds.

Step 1: Plot the Graph

Begin by creating the x and y axes. The x-axis will represent time \(t\) in seconds, and the y-axis will represent speed \(S\) in feet per second. Then plot the data points provided in the table. Join these points to form the line graph.

Step 2: Analyze the Graph

From the graph, observe how the speed of the object changes with time. Analyze whether there is a certain speed at which the object seems to remain stable, indicating a possible limiting speed.

Step 3: Identify the Limiting Speed and Possible Cause.

If the graph tends to flatten at a certain speed \(S\), it can be concluded that this is the limiting speed. A possible cause may be air resistance, which tends to balance the force of gravity when an object reaches a certain speed, preventing the object from accelerating further.