In 2005, Hurricane Rita hit several states in the southern United States. In the panic to escape her wrath, thousands of people tried to flee Houston, Texas by car. One car full of college students traveling to Tyler, Texas, 199 miles north of Houston, moved at an average speed of \(3.0 \mathrm{~m} / \mathrm{s}\) for one-fourth of the time, then at \(4.5 \mathrm{~m} / \mathrm{s}\) for another one-fourth of the time, and at \(6.0 \mathrm{~m} / \mathrm{s}\) for the remainder of the trip. a) How long did it take the students to reach their destination? b) Sketch a graph of position versus time for the trip.

Understanding the concept of average speed is crucial for interpreting motion in physics. Average speed is defined as the total distance traveled divided by the total time taken to cover that distance. In mathematical terms, the average speed, V_avg, can be represented by the equation:
\[\begin{equation} V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} \end{equation}\]
Consider our example with the students fleeing Hurricane Rita. Here, we want to find out how long it took to travel 199 miles. The average speed isn't constant, but changes in a piecewise manner as they traveled at different speeds during different portions of their journey. To calculate the total time of travel, one approach is to determine the distance covered in each segment at the given speeds and then sum these distances before converting to total time using the average speed equation above. This is an essential skill in kinematics, as it allows one to analyze motion over varying speed segments and still determine overall time for travel.

A distance-time graph is an invaluable tool in kinematics, enabling us to visualize an object's motion over time. Each point on the graph represents the position of an object at a specific time. The steeper the graph's slope, the faster the object is moving. Conversely, a flat line indicates the object is stationary.
To construct a distance-time graph for our hurricane evacuation scenario, you would plot time on the horizontal axis and distance on the vertical axis. For each speed segment, you would calculate the distance using the average speed and the time taken. After plotting these calculated distances against the corresponding times, straight lines would connect these points, thereby creating a piecewise linear graph. The slope of each segment corresponds to the speed at that particular part of the trip, depicting the changes in speed visually as the students traveled to their destination.

In physics problems, especially those involving kinematics, it's imperative to check that all measurements are in compatible units before beginning any calculations. Converting units is a common and necessary step for working through many physics exercises.
For instance, in the problem concerning the students escaping Hurricane Rita, distances are initially given in miles, while speeds are in meters per second. To proceed with the calculations, we must convert miles to meters, since meters per second is the standard unit for speed in the International System of Units (SI). One mile is equivalent to 1609.34 meters. By multiplying the total distance in miles by this conversion factor, we obtain the distance in meters, allowing for coherent and accurate calculations. Always remember, correctly converting units can make the difference between a right and wrong answer in physics problems.