Calculate Andrea's Speed On A Circular Track A Relative Motion Problem
This article dives into a classic mathematical problem involving relative motion and explores how to determine the speeds of individuals running in opposite and the same directions around a circular track. Understanding these concepts is fundamental not only in mathematics but also in physics and various real-world applications. We'll break down the problem step-by-step, focusing on clarity and providing a comprehensive guide for readers of all backgrounds. Our goal is to provide you with the knowledge and skills to tackle similar problems with confidence. This article aims to clarify the core concepts of relative speed and how it applies to scenarios where individuals are moving in circular paths, either towards each other or in the same direction. By the end of this discussion, you will understand how to use the provided information—the circumference of the track, the time it takes for the runners to meet when moving in opposite directions, and the time it takes for them to meet when moving in the same direction—to calculate the individual speeds of the runners.
Imagine two runners, let's call them Andrea and another person, are practicing on a circular track. The track's circumference is 150 meters. When Andrea and the other runner sprint in opposite directions, they meet every 5 seconds. However, when they run in the same direction from the same starting point, they are together again every 25 seconds. Our objective is to calculate Andrea's speed. This problem is a fantastic example of how relative speeds change depending on the direction of motion. The key to solving this lies in understanding how the combined speeds affect the time it takes for the runners to meet or be in the same position again. The problem challenges us to think critically about how motion is perceived in different frames of reference and to use this understanding to calculate individual speeds. This kind of problem is not just a mathematical exercise; it's a way to sharpen our logical reasoning and problem-solving abilities.
Relative speed is the speed of an object with respect to another. It's a crucial concept when dealing with scenarios where multiple objects are in motion. When objects move in opposite directions, their speeds add up, making their relative speed the sum of their individual speeds. This is because they are effectively covering the distance between them at a faster rate. Conversely, when objects move in the same direction, their relative speed is the difference between their individual speeds. The faster object is essentially catching up to the slower one, so the relative speed is lower. Understanding this distinction is key to solving problems like the one presented, where the same runners behave differently depending on their directions of movement. Relative speed is not just a theoretical concept; it has practical applications in many areas, from navigation and aviation to everyday scenarios like driving a car on the highway. For instance, when overtaking another vehicle, understanding relative speed helps in making safe and efficient maneuvers.
Opposite Directions
When Andrea and the other runner move in opposite directions, their speeds combine. This means that the distance between them closes much faster than if they were each running in isolation. To visualize this, imagine two cars moving towards each other on a road; the rate at which they approach each other is the sum of their speeds. In our track scenario, the combined speed allows them to cover the entire circumference of the track (150 meters) in just 5 seconds. This faster closing speed is why they meet so quickly. Mathematically, if Andrea's speed is A and the other runner's speed is B, their relative speed when moving in opposite directions is A + B. This concept is fundamental to understanding how relative motion works and is crucial for setting up the equations needed to solve the problem. Understanding how speeds combine in opposite directions helps us to appreciate the dynamics of motion from different perspectives.
Same Direction
When Andrea and the other runner are running in the same direction, the dynamics change significantly. Instead of their speeds adding up, we now consider the difference in their speeds. This is because the faster runner is essentially chasing the slower runner around the track. The relative speed in this case determines how quickly the faster runner gains a full lap on the slower runner. For them to be together again, the faster runner must cover the entire circumference of the track more than the slower runner. This takes longer, as the relative speed is lower. If Andrea is the faster runner, the relative speed when running in the same direction is A - B (assuming A is greater than B). This slower relative speed is why it takes 25 seconds for them to meet again when running in the same direction, compared to just 5 seconds when running in opposite directions. The concept of relative speed in the same direction is essential for understanding how overtaking maneuvers work in various real-life scenarios.
To solve this problem, we need to translate the given information into mathematical equations. We have two scenarios: runners moving in opposite directions and runners moving in the same direction. Let's denote Andrea's speed as A (in meters per second) and the other runner's speed as B (in meters per second). The circumference of the track is 150 meters.
Equation 1: Opposite Directions
When running in opposite directions, their relative speed is A + B. They meet every 5 seconds, which means that in 5 seconds, they cover the entire circumference of the track. This gives us our first equation:
5(A + B) = 150
This equation tells us that the combined distance they run in 5 seconds equals the total distance around the track. It's a direct application of the principle that distance equals speed multiplied by time. Setting up this equation is a crucial step in solving the problem, as it captures the essence of their motion when moving towards each other.
Equation 2: Same Direction
When running in the same direction, their relative speed is |A - B| (the absolute value ensures we're dealing with a positive speed difference). They are together every 25 seconds, meaning the faster runner has gained a full lap (150 meters) on the slower runner in that time. This gives us our second equation:
25|A - B| = 150
This equation reflects the scenario where the faster runner needs to complete an additional lap to catch up with the slower runner. The use of the absolute value is important because we don't initially know which runner is faster. This equation is vital for understanding how the difference in speeds affects the time it takes for the runners to meet when moving in the same direction.
Now that we have our two equations, we can solve for A and B. Let's start by simplifying both equations:
Simplifying Equation 1
Divide both sides of the equation 5(A + B) = 150 by 5:
A + B = 30
This simplified equation gives us a direct relationship between the speeds of the two runners. It tells us that their combined speed is 30 meters per second. This simplification makes the next steps in solving the system of equations much easier.
Simplifying Equation 2
Divide both sides of the equation 25|A - B| = 150 by 25:
|A - B| = 6
This simplified equation tells us that the difference in their speeds is 6 meters per second. It's important to note the absolute value, which means we have two possibilities: A - B = 6 or B - A = 6. However, for the context of this problem, we'll assume A > B (Andrea is faster), so we'll proceed with A - B = 6. If we assumed the opposite, the final result would still be the same, but the intermediate steps would involve a bit more algebraic manipulation.
Solving for A and B
We now have a system of two linear equations:
- A + B = 30
- A - B = 6
We can solve this system using various methods, such as substitution or elimination. Let's use the elimination method. Add the two equations together:
(A + B) + (A - B) = 30 + 6
2A = 36
Now, divide both sides by 2 to solve for A:
A = 18
So, Andrea's speed is 18 meters per second. Now we can substitute this value back into either equation to solve for B. Let's use the first equation:
18 + B = 30
Subtract 18 from both sides:
B = 12
Therefore, the other runner's speed is 12 meters per second. We have successfully solved for both A and B, giving us a complete understanding of the runners' speeds.
Andrea's speed is 18 meters per second.
This problem demonstrates the power of understanding relative motion and how it affects the interaction between moving objects. By carefully setting up equations based on the given information, we were able to determine Andrea's speed. The key takeaways from this problem are the concepts of relative speed in opposite and the same directions, and how these concepts translate into mathematical equations. Problems like these are not just academic exercises; they help develop critical thinking and problem-solving skills that are valuable in many areas of life. Understanding relative motion is crucial in various fields, from transportation and logistics to sports and even astronomy. By mastering these concepts, we can gain a deeper understanding of the world around us and how objects interact within it.
This comprehensive guide has walked you through the problem step-by-step, providing insights into the underlying principles and methods used to solve it. With this knowledge, you should be well-equipped to tackle similar problems involving relative motion and circular tracks. Remember, the key is to break down the problem into manageable parts, understand the physics involved, and translate that understanding into mathematical equations. Practice makes perfect, so try solving similar problems to reinforce your understanding and build your confidence.