In 2005, Hurricane Rita hit several states in the southern...

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.

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.

Key Concepts

Piecewise Analysis of Motion

Piecewise analysis of motion is the technique of breaking down a complex motion into segments in which the motion is uniform or follows a simple pattern. Each segment is analyzed separately using the uniform motion principles, and then the results are combined to give a complete description of the overall journey. This method is particularly useful when speeds or directions change at certain intervals during a trip.

Time Partitioning

Time partitioning involves dividing the total time of a journey into segments based on given proportions or intervals. In problems where different speeds apply for different parts of the trip, understanding and assigning the correct fraction of the total time to each segment is essential for accurately calculating the distances covered during each phase of the motion.

Graphical Representation of Motion

A graphical representation of motion, such as a position versus time graph, provides a visual depiction of how an object's position changes over time. In the context of piecewise uniform motion, the graph typically consists of linear segments with slopes proportional to the speeds during each interval. Analyzing these graphs helps in understanding the relationship between speed, time, and distance, and in identifying key features like changes in speed or direction.

Uniform Motion

Uniform motion refers to motion at a constant speed in a straight line. In this concept, the distance traveled is directly proportional to the time elapsed, which simplifies calculations by allowing the use of the relation distance = speed × time. This principle is fundamental in kinematics and helps analyze each segment of motion where the speed is constant.

Distance Calculation

Distance calculation involves determining the distance traveled by using the relationship distance = speed × time. When different segments of a trip occur at different constant speeds, the total distance is the sum of distances from each segment. This approach is crucial when dealing with piecewise motion, where each segment can be analyzed independently before combining the results.

Transcript

0:00 Hello.

00:02 So in this problem, we're speaking about a hurricane that happened in 2005.

00:08 And due to this, some students flee from houston to texas, which is 199 meters north from houston.

00:19 The description of the trip that the students traveled the first quarter of their trip at 3 meters per second, then the next quarter they travel at 4 .5 meters per second, and for the rest of the trip, which is half of time, they were traveling at 6 meters per second.

00:46 The question is to determine how long did it take for those students to reach their destination at texas? so in order to do this, let's first determine the distance from houston to texas but in meters instead of mines.

01:03 So we know that one mile is equal to 1 ,609 meters approximately.

01:13 So the distance in that case is 199 multiplied by 1009, which is approximately 3009.

01:27 120 times 10 to the power of 3 meters.

01:32 So that's the distance measured in meters.

01:37 In order to get the time, we need to get the average velocity.

01:42 And if we divide the distance by the average velocity, now we have the time that it takes for the students to get there.

01:52 So to get the average velocity, the average, we need here to multiply 3.

02:04 Times one quarter plus 4 .5 multiplied by one quarter plus 6 multiplied by one half.

02:17 And we will get the average velocity which is 4 .875.

02:24 Then we can directly calculate the time by dividing distance over velocity or the average velocity.

02:35 So here we have 320.

02:38 Multiplied by 10 to the power of 3 divided by 4 .875.

02:47 And the answer is around 65 ,0694.

02:58 Approximately, this is in seconds.

03:01 Dividing this by 3 ,600, we get the answer which is around 18 .2 hours.

03:08 And that's the time that it takes the students to reach their destination.

03:16 In the second part, we're asked to sketch a graph of the position versus time for this entire trip...

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