Impossibly Fast Speed Required for a Train to Achieve a Total Average Speed
Calculating the speed required for the second leg of a train's journey is a fascinating exercise in physics. Let's break down the problem step by step, and see just how unrealistic the speeds become.
Understanding the Initial Conditions
The train initially travels the first leg of its trip at a speed of 17 m/s, taking 3.3 hours. To better understand these figures, let's convert the speed and time into more familiar units of kilometers per hour (km/h).
Conversions and Calculations
1. **Convert the initial speed to km/h:**
The initial speed is 17 m/s. To convert this to km/h, we multiply by 3.6 (since 1 m/s 3.6 km/h). Therefore, the initial speed is approximately 61.2 km/h.
2. **Calculate the first leg's travel time in hours and minutes:**
3.3 hours is already in the desired unit, but it's helpful to convert this to minutes as well.
3.3 hours 3 hours 18 minutes.
3. **Confirm the distance of the first leg:**
The distance covered in the first leg is calculated as:
Distance time × velocity 3.3 hours × 61.2 km/h ≈ 201.96 km.
Setting the Target Average Speed
The goal is for the train to achieve a total average speed of 30 m/s (108 km/h) over the entire 400 km journey. To understand this, let's first determine the total time required to travel this distance:
The Total Time Calculation
1. **Total distance and time required to achieve the target speed:**
The total distance is 400 km, and with a target average speed of 108 km/h, the total time is:
Total time total distance / average speed 400 km / 108 km/h ≈ 3.70 hours (3 hours 42 minutes).
2. **Determine the remaining travel time for the second leg:**
The first leg took approximately 3.3 hours, so the remaining time for the second leg is:
Remaining time total time - first leg time 3.70 hours - 3.3 hours 0.40 hours (24 minutes).
The Impossibility of the Second Leg Distance
With the remaining time and the total distance for the second leg, we can calculate the required speed:
Second Leg Speed Calculation
1. **Second leg distance:**
The second leg of the journey is 200 km.
2. **Convert the remaining time to seconds for accuracy:**
0.40 hours 0.40 × 3600 seconds 1440 seconds (since 1 hour 3600 seconds).
3. **Calculate the required speed for the second leg:**
Required speed remaining distance / remaining time 200 km / 1440 s ≈ 138.89 m/s (or 499.01 km/h).
Given that 499.01 km/h is approximately 310 mph, it's clear that this speed is extraordinarily high and essentially impossible for any conventional train to achieve.
Conclusion
While the calculations show a theoretical possibility, the practical reality is that achieving such a high speed would be impossible with current technology and infrastructure. Therefore, the problem represents a theoretical exercise rather than a practical one.