Which best describes the runners?
A. Garreth and Elisa ran the same distance but not the same displacement, Luigi and Jasmine went the same displacement, and Logan ran faster than Jasmine.
B. Garreth ran farther than Luigi, Elisa and Jasmine went the same distance, and Logan and Jasmine have the same displacement in the opposite direction.
C. Garreth and Elisa ran together, Luigi and Jasmine went the same distance, and Logan and Jasmine have the same displacement in the opposite direction.
D. Garreth ran less than Logan, Elisa and Jasmine went the same displacement, and Luigi and Garreth ran together.
Answer (2)
Assign names to the runners based on their velocities: Garreth (5.5 mph west), Elisa (4.3 mph east), Luigi (5.5 mph west), Jasmine (4.0 mph south), Logan (4.0 mph north).
Analyze each statement to determine its accuracy based on the runners' velocities and directions.
The second statement, "Garreth ran farther than Luigi, Elisa and Jasmir went the same distance, and Logan and Jasmir has the same displacement in the opposite direction," best describes the runners, assuming Garreth ran longer than Luigi and Elisa and Jasmine ran for different times to cover different distances.
Therefore, the best description of the runners is the second statement.
Explanation
Assigning Names to Runners We are given the velocities of five runners and four statements describing their relationships. Our goal is to determine which statement best describes the runners. Let's assign names to the runners based on their velocities: Garreth (5.5 mph west), Elisa (4.3 mph east), Luigi (5.5 mph west), Jasmine (4.0 mph south), Logan (4.0 mph north).
Analyzing the First Statement Let's analyze the first statement: Garreth and Elisa ran the same distance but no same displacement, Luigi and Jasmine went the same displacement, and Logan ran faster than Jasmine. Compare Garreth's speed (5.5 mph) and Elisa's speed (4.3 mph). Since their speeds are different, for them to run the same distance, they would have to run for different amounts of time. So, it's possible for them to run the same distance. Garreth runs west and Elisa runs east, so their displacements are different. Luigi's velocity is 5.5 mph west, and Jasmine's velocity is 4.0 mph south. Their displacements are not the same. Logan's speed is 4.0 mph and Jasmine's speed is 4.0 mph, so Logan did not run faster than Jasmine. Therefore, the first statement is not entirely correct.
Analyzing the Second Statement Let's analyze the second statement: Garreth ran farther than Luigi, Elisa and Jasmine went the same distance, and Logan and Jasmine has the same displacement in the opposite direction. Garreth and Luigi have the same speed (5.5 mph). If they ran for the same amount of time, they ran the same distance. If Garreth ran for a longer time, he ran farther. Elisa's speed is 4.3 mph and Jasmine's speed is 4.0 mph. If they ran for the same amount of time, they did not run the same distance. Logan's velocity is 4.0 mph north and Jasmine's velocity is 4.0 mph south. If they ran for the same amount of time, their displacements are the same in the opposite direction. This statement could be true if Garreth ran longer than Luigi and Elisa and Jasmine ran for different times to cover different distances.
Analyzing the Third Statement Let's analyze the third statement: Garreth and Elisa ran together, Luigi and Jasmine went the same distance, and Logan and Jasmine have the same displacement in the opposite direction. Garreth's velocity is 5.5 mph west and Elisa's velocity is 4.3 mph east. They did not run together. Luigi's velocity is 5.5 mph west and Jasmine's velocity is 4.0 mph south. They did not run the same distance if they ran for the same time. Logan's velocity is 4.0 mph north and Jasmine's velocity is 4.0 mph south. If they ran for the same amount of time, their displacements are the same in the opposite direction. This statement is incorrect because Garreth and Elisa did not run together and Luigi and Jasmine did not run the same distance.
Analyzing the Fourth Statement Let's analyze the fourth statement: Garreth ran less than Logan, Elisa and Jasmine went the same displacement, and Luigi and Garreth ran together. Garreth's speed is 5.5 mph and Logan's speed is 4.0 mph. If they ran for the same amount of time, Garreth ran farther than Logan. Elisa's velocity is 4.3 mph east and Jasmine's velocity is 4.0 mph south. Their displacements are not the same. Luigi and Garreth have the same velocity, so they ran together. This statement is incorrect because Garreth did not run less than Logan and Elisa and Jasmine did not have the same displacement.
Conclusion Comparing the analysis of each statement, the second statement best describes the runners, assuming Garreth ran longer than Luigi and Elisa and Jasmine ran for different times to cover different distances.
Examples
Understanding relative motion and displacement is crucial in many real-world scenarios, such as air traffic control, where controllers must track the positions and velocities of multiple aircraft to ensure safe separation. Similarly, in maritime navigation, captains need to account for the velocities of their ships and other vessels, as well as the effects of currents and winds, to avoid collisions and reach their destinations efficiently. In sports, athletes and coaches use these concepts to analyze performance, optimize strategies, and make informed decisions, such as determining the best angle and velocity for a jump or throw.
Statement B best describes the runners, as it accurately reflects their distances and displacements given that Garreth and Luigi can run the same distance, and Logan and Jasmine have equal displacements in opposite directions. The other statements either contain inaccuracies regarding distances or speeds. Therefore, the correct option is B.
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