A train travels 120 km at a speed of 80 km/h, then another 180 km at a speed of 90 km/h. What is the average speed of the train for the entire journey?
["Average Speed of a Train Journey: Understanding Travel Efficiency", "When analyzing train travel, one common question arises: What is the average speed of a train over successive segments with different speeds? In this example, a train travels 120 kilometers at 80 km/h, then another 180 kilometers at 90 km/h. At first glance, the mean of the two speeds (85 km/h) might come to mind—but this isn’t the correct way to calculate average speed for variable journeys. Instead, average speed depends on total distance and total time. Let’s explore how to compute it accurately.", "---", "### The Journey Breakdown", "The train travels two distinct legs:", "1. First leg:\n Distance: 120 km\n Speed: 80 km/h\n Time = Distance ÷ Speed = 120 km ÷ 80 km/h = 1.5 hours", "2. Second leg:\n Distance: 180 km\n Speed: 90 km/h\n Time = 180 km ÷ 90 km/h = 2 hours", "---", "### Total Distance and Total Time", "- Total distance = 120 km + 180 km = 300 km\n- Total time = 1.5 hours + 2 hours = 3.5 hours", "---", "### How to Calculate Average Speed", "The formula for average speed is:", "[\n\ ext{Average Speed} = \frac{\ ext{Total Distance}}{\ ext{Total Time}}\n]", "Plugging in the values:", "[\n\ ext{Average Speed} = \frac{300 \ ext{ km}}{3.5 \ ext{ h}} \approx 85.71 \ ext{ km/h}\n]", "---", "### Why the Simple Average (80 + 90 ÷ 2 = 85 km/h) Falls Short", "Using the arithmetic mean of the two speeds:\n[\n\frac{80 + 90}{2} = 85 \ ext{ km/h}\n]", "While intuitive, this method underestimates the true average speed because it doesn’t account for the time spent at each speed. The slower segment covers a longer share of the total time, thus pulling the average down. The correct calculation considers duration, not just rates.", "---", "### Real-World Implications", "Understanding average speed is crucial for scheduling, safety, and efficiency in rail transport. Engineers and planners rely on precise average speed calculations to optimize routes, manage timetables, and assess fuel or energy consumption. In urban rail systems — such as commuter trains or high-speed lines — knowing the true average speed helps operators communicate realistic journey times to passengers.", "---", "### Summary", "- Total distance: 300 km\n- Total time: 3.5 hours\n- Correct average speed: ~85.71 km/h", "Remember: when旅行 judged by distance and time over multiple segments, always divide total distance by total time — not by the arithmetic mean of individual speeds. This classic physics principle ensures accurate travel insights for trains, cars, and all transportation modes.", "---", "Keywords: average speed train, calculate average speed, train travel time calculation, train speed efficiency, total distance divided by total time, real-world route planning, train journey analysis, speed and distance formula."]
