Problem 62 - College Algebra with Modeling and Visualization

Question

Sketch a graph that illustrates the motion of the person described. Let the \(x\) -axis represent time and the \(y\) axis represent distance from home. Be sure to label each axis. A person drives to a nearby park at 25 miles per hour for 1 hour, rests at the park for 2 hours, and then drives home at 50 miles per hour.

Step-by-Step Solution

Answer

A two-segment slope upwards, flat period, then a steeper slope downwards.

Step 1: Set Up the Axes

To begin sketching, label the horizontal axis as the time axis ( t ) and the vertical axis as the distance from home axis ( d ). Each unit on the x -axis will represent 1 hour, while each unit on the y -axis will represent 25 miles, since that will help easily plot the speeds mentioned.

Step 2: Sketch the Drive to the Park

The person drives at 25 miles per hour for 1 hour. Plot a line starting from the origin (0,0) and extending to the point (1,25) on the graph, representing the distance covered in that time.

Step 3: Plot the Resting Period

For the next 2 hours, the person rests at the park. Therefore, draw a horizontal line from the point (1,25) to (3,25) to indicate there is no change in the distance from home during this period.

Step 4: Sketch the Drive Home

Now, the person drives home at 50 miles per hour. From the point (3,25) draw a line that slopes downward. This line should reach the horizontal axis again at (3.5,0), as the return trip covers 25 miles in just 0.5 hours.

Key Concepts

Distance-Time Graph

A distance-time graph is a powerful tool that helps you visualize the distance traveled over a period by plotting it on a graph. The horizontal axis (the x-axis) represents time, and the vertical axis (the y-axis) represents distance from a specific starting point. In this case, we are considering a scenario where a person drives to a park and back home. On a distance-time graph:

• A straight upward sloping line indicates uniform motion, meaning the traveler is moving at a constant speed.
• A horizontal line suggests the person is at rest, with no change in distance over time.
• A straight downward sloping line, like the one in the return trip, indicates a change in direction or returning towards the starting point.

By interpreting the slopes and shapes of lines, you can understand different phases of the journey, such as accelerating, resting, or returning.

Motion Representation

The motion of an object, like a car on a road, can be effectively represented on a graph, providing valuable insights into its journey. In the described exercise, the graph represents different stages of motion. Initially, the graph shows an upward slope, illustrating the person driving to the park at a constant speed of 25 miles per hour. This upward slope indicates that the distance from home increases consistently.
The flat line which follows represents the rest period at the park, showing no change in the distance during this time. Following the resting period, the graph’s downward slope captures the high-speed return trip at 50 miles per hour, showing that in a short duration, the person returns to the starting point.

• This allows us to observe the constant speed, resting state, and return home all in one glance.
• Looking at the slope angles and transitions helps us understand how speeds vary during the journey.

Using distance-time graphs for motion representation helps students link physical concepts, like speed and rest, to graphical elements in a simple, visual manner.

Graphical Analysis

Graphical analysis involves examining the plotted points and lines on a graph to derive meaningful interpretations. This skill is crucial for understanding motion as it presents complex data simply.
Let's break down the distance-time graph in this scenario. The initial upward line segment (from 0 to 1) is steep and straight, demonstrating constant motion towards the park for one hour at 25 miles per hour — each increment on the graph clearly reflects these units.
Next, the horizontal line (from 1 to 3) indicates a rest period with no additional distance covered.

• This shows how a lack of slope signifies stillness, making it a clear visual metaphor for being stopped.

Lastly, the steeper downward line (from 3 to 3.5) signifies the return journey at a faster speed of 50 miles per hour, evidenced by how rapidly it descends back to the starting point.
Graphical analysis allows us to not just check distances traveled, but also infer and understand the speed and vital periods of rest in an easy to digest manner.