Calculating Distance Between Vehicles A Physics Problem Solved!

Hey guys! Let's dive into a cool physics problem that involves calculating the distance between two vehicles moving at different speeds. This is a classic scenario that helps us understand the relationship between speed, time, and distance. We'll break down the problem step by step, so you can easily grasp the concepts and apply them to similar situations.

Understanding the Problem

So, here’s the situation: We have two vehicles, each cruising along at different speeds – one at 10 km per hour and the other at 12 km per hour. They're on a collision course, but not in a destructive way! They're actually moving perpendicularly to each other. Think of it like one car heading north and the other heading east. They cross paths, and we want to figure out how far apart they are after six hours of traveling. This problem is a fantastic way to see how distances accumulate over time when objects are moving at constant speeds in different directions. It's also a practical application of the Pythagorean theorem, which we'll get into later. Understanding the speeds is crucial; one vehicle is slightly faster, meaning it will cover more ground in the same amount of time. This difference in distance traveled is key to solving the problem. The time frame of six hours is also a critical piece of information. It gives us the duration over which these vehicles are traveling, allowing us to calculate the total distance each one covers. Visualizing the problem as two vehicles moving at right angles to each other is super helpful. This setup forms a right triangle, where the distances traveled by each vehicle are the legs, and the distance between them is the hypotenuse. Recognizing this geometric relationship is essential for applying the Pythagorean theorem correctly. So, let's put on our thinking caps and get ready to solve this real-world physics puzzle!

Calculating the Distances Traveled

The first step in solving this problem is to figure out how far each vehicle travels in those six hours. Remember the basic formula: distance = speed × time. This formula is the bread and butter of these types of problems, so make sure you've got it down! For the first vehicle, which is traveling at 10 km per hour, the calculation is straightforward. We multiply its speed by the time it travels: 10 km/hour × 6 hours = 60 km. So, this vehicle covers 60 kilometers in six hours. Now, let's do the same for the second vehicle. It’s moving a bit faster at 12 km per hour. Using the same formula, we get: 12 km/hour × 6 hours = 72 km. This vehicle travels 72 kilometers in the same amount of time. Notice that because the second vehicle is faster, it covers a greater distance in the six-hour period. This difference in distance is important because it will affect the final distance between the two vehicles. These calculations give us the lengths of the two legs of our right triangle. One leg is 60 km, and the other is 72 km. With these distances in hand, we're one step closer to finding the hypotenuse, which represents the distance between the vehicles. It's like we're building the foundation of our solution, piece by piece. We've taken the given information – the speeds and the time – and transformed it into the actual distances each vehicle has covered. This is a crucial step because we can now visualize the problem in terms of spatial distances rather than just speeds and time. So, we've got the distances covered by each vehicle. What's next? We're going to use these distances to find the final distance between the vehicles, and that's where the Pythagorean theorem comes in!

Applying the Pythagorean Theorem

Okay, guys, now we're getting to the fun part! We're going to use the Pythagorean theorem to find the distance between the two vehicles. Remember that this theorem applies to right triangles and states that a² + b² = c², where 'a' and 'b' are the lengths of the legs (the distances each vehicle traveled), and 'c' is the length of the hypotenuse (the distance between the vehicles). So, in our case, 'a' is 60 km (the distance traveled by the first vehicle), and 'b' is 72 km (the distance traveled by the second vehicle). We want to find 'c', which is the distance separating them. Let's plug the values into the formula: 60² + 72² = c². First, we need to calculate the squares: 60² = 3600 and 72² = 5184. Now, add these together: 3600 + 5184 = 8784. So, we have 8784 = c². To find 'c', we need to take the square root of 8784. Using a calculator (or some good old-fashioned math skills!), we find that the square root of 8784 is approximately 93.72 km. This means that after six hours, the two vehicles are about 93.72 kilometers apart. Isn't that cool? We've used a fundamental physics concept and a bit of geometry to solve a real-world problem. The Pythagorean theorem is such a powerful tool in situations like this, where we have objects moving at right angles to each other. It allows us to bridge the gap between the individual distances traveled and the overall separation. This step is like putting the final piece in a puzzle. We've used the distances we calculated earlier and the Pythagorean theorem to arrive at our answer. It's a testament to how mathematical principles can be applied to understand and solve problems in the physical world.

Final Answer and Summary

Alright, let's wrap things up and state our final answer. After six hours of traveling, the two vehicles are approximately 93.72 kilometers apart. We did it! We successfully calculated the distance using the concepts of speed, time, distance, and the Pythagorean theorem. This problem demonstrates a beautiful blend of physics and mathematics. We started by understanding the scenario: two vehicles moving at different speeds, perpendicularly to each other. Then, we calculated the individual distances each vehicle traveled using the formula distance = speed × time. Finally, we applied the Pythagorean theorem to find the distance between the vehicles, recognizing that the problem formed a right triangle. This type of problem is a great example of how physics can be used to model real-world situations. It shows how we can use simple principles to predict the outcome of more complex scenarios. Understanding these concepts isn't just about solving problems in a textbook; it's about developing a way of thinking that allows you to analyze and understand the world around you. So, next time you see objects moving at different speeds and directions, remember this problem. Think about how you can break it down into simpler components and use the tools of physics and mathematics to find a solution. And remember, the key is to stay curious, keep practicing, and never stop asking questions!

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What is the distance separating two vehicles after six hours if they travel perpendicularly at 10 km/h and 12 km/h respectively?

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Calculating Distance Between Vehicles A Physics Problem Solved!