Calculating Average Speed: A Cyclist's Journey

Hey guys! Let's dive into a classic physics problem: calculating the average speed of a cyclist over different segments of their journey. This is a super common type of question, and understanding how to solve it is a great way to solidify your understanding of speed, distance, and time. We'll break down the problem step-by-step, making it easy to follow along. So, grab your calculators, and let's get started! We'll cover both the average speed for the entire trip and the average speed for just the first half of the distance.

Decoding the Problem

The problem states: "A cyclist covered 10 km in the first 0.5 hours, then traveled at 25 km/h for the next 12 minutes. The remaining 9 km were covered at a speed of 18 km/h. What is the cyclist's average speed: a) over the entire path? b) in the first half of the path?" Sounds straightforward, right? But like any good physics problem, we need to carefully break down the information to avoid any confusion. Pay attention, because we will need to calculate the total distance traveled by the cyclist, as well as the total time spent on each segment of the journey. Keep in mind that speed = distance / time. Therefore, we'll need to know these values for each segment to figure out the overall average speeds. We'll also have to deal with different units (hours and minutes), so unit conversion is key here. Make sure we're consistent with our units; using kilometers and hours will keep things simple. Keep in mind that we need to find the total distance and total time for the entire journey and the first half of the journey to solve the problems. Finally, remember that average speed is NOT just the average of the speeds. It's the total distance divided by the total time. Let’s get to it!

Solving for the Whole Trip (Part A)

Okay, let's tackle part (a): finding the average speed for the entire journey. This requires us to know the total distance traveled and the total time taken. We've got the info, but we need to do some calculations and conversions first! We already know the distance for the first and third segments (10 km and 9 km, respectively). Now, we need to figure out the distance traveled during the second segment and the time spent in the third segment. For the second segment, we know the speed (25 km/h) and the time (12 minutes). First things first, we must convert those 12 minutes to hours. This is an important step; if you forget this, the math will be way off. There are 60 minutes in an hour, so 12 minutes is equal to 12/60 = 0.2 hours. Now we can calculate the distance for the second segment: distance = speed * time = 25 km/h * 0.2 h = 5 km. Sweet! Now, let’s find the time it took to travel the third segment. We know the distance is 9 km and the speed is 18 km/h. Rearranging the speed formula (speed = distance/time), we get time = distance/speed. So, the time for the third segment is 9 km / 18 km/h = 0.5 hours.

Now, let's gather all the information.

• Segment 1: Distance = 10 km, Time = 0.5 h
• Segment 2: Distance = 5 km, Time = 0.2 h
• Segment 3: Distance = 9 km, Time = 0.5 h

Next, to find the total distance, we add up the distances of each segment: 10 km + 5 km + 9 km = 24 km. The total time is the sum of the times for each segment: 0.5 h + 0.2 h + 0.5 h = 1.2 h. Finally, we can calculate the average speed for the entire journey: average speed = total distance / total time = 24 km / 1.2 h = 20 km/h. So, the average speed of the cyclist over the entire path is 20 km/h. Awesome! We've successfully completed the first part of the problem.

Finding the Average Speed for the First Half of the Path (Part B)

Alright, let's move on to part (b), where we have to find the average speed of the cyclist in the first half of the path. This requires a slightly different approach because, instead of using all the distances, we must determine when the cyclist has covered half the total distance. We already know the total distance is 24 km. So, the first half of the path is 24 km / 2 = 12 km. This is important! The trick here is to figure out at what point the cyclist reaches 12 km. The first segment covers 10 km. So we need to determine how much of the second segment the cyclist needs to complete the first half. We know the second segment is 5km long. Since the first half is 12 km, and the first segment is 10 km, then we just need 2 more km from the second segment.

We know the cyclist travels at 25 km/h in the second segment. To calculate the time it takes to cover 2 km, we use the formula time = distance/speed. Therefore, the time is 2 km / 25 km/h = 0.08 hours. Again, remember that the first segment took 0.5 hours. So, the total time for the first half of the journey is 0.5 hours (segment 1) + 0.08 hours (part of segment 2) = 0.58 hours. Now that we have the distance (12 km) and the time (0.58 hours) for the first half of the path, we can calculate the average speed: average speed = total distance / total time = 12 km / 0.58 h ≈ 20.69 km/h. Thus, the average speed of the cyclist in the first half of the path is approximately 20.69 km/h. Boom! You've got it, guys!

Summary and Key Takeaways

In this problem, we walked through the process of calculating average speed in multiple stages. Here's a recap of what we covered:

• We first determined the average speed for the entire trip, finding the total distance and the total time. Remember to convert units! This involved breaking the journey down into segments and calculating the time or distance for each segment as needed.
• Then, we found the average speed for the first half of the path. This involved identifying the halfway point in terms of distance, determining how long it took to reach that point, and then calculating the average speed for that portion of the journey.
• The most important thing to remember is the formula: Average Speed = Total Distance / Total Time. Don't fall for just averaging the speeds! Always consider the total distance covered and the total time taken.

By following these steps, you can confidently tackle similar problems. Keep practicing, and you'll become a pro at these calculations. The key is breaking down the problem into smaller parts, understanding the relationships between speed, distance, and time, and paying close attention to units. Keep up the awesome work!