Calculating Average Velocity in a Real-life Scenario: A Cyclist's Journey
Imagine a cyclist embarking on a journey to a nearby town, over a period of 70 minutes. This adventurous trip involves several distinct segments, each with varying speeds and durations. Understanding how to calculate the average velocity of the entire trip is both a practical and academic pursuit.
The Context of the Trip
The cyclist's journey can be divided into four segments as follows:
First Segment: 30 minutes at a speed of 30 km/h. Second Segment: 15 minutes at a speed of 40 km/h. Third Segment: 5 minutes at a speed of 0 km/h for a break. Fourth Segment: 20 minutes at a speed of 15 km/h.Calculating Individual Distances
To begin, let us calculate the distance for each segment separately:
First Segment: Distance 30 km/h times; (30/60) hours 15 km Second Segment: Distance 40 km/h times; (15/60) hours 10 km Third Segment: No movement, so distance 0 km Fourth Segment: Distance 15 km/h times; (20/60) hours 5 kmTotal distance 15 km 10 km 0 km 5 km 30 km
Calculating Total Time
The total time of the trip is:
30 minutes 15 minutes 5 minutes 20 minutes 70 minutes
Converting this to hours for the average velocity calculation:
70 minutes ÷ 60 1.1667 hours
Calculating Average Velocity
Using the formula for average velocity, which is total distance divided by total time:
Average velocity 30 km ÷ 1.1667 hours 25.7143 km/h
Alternatively, using the mathematical expression as follows:
[frac{30 times frac{30}{60} 40 times frac{15}{60} 15 times frac{20}{60}}{frac{30}{60} frac{15}{60} frac{5}{60} frac{20}{60}}]
Simplifying the numerator:
[15 10 5 30]
Simplifying the denominator:
[frac{30}{60} frac{15}{60} frac{5}{60} frac{20}{60} frac{30 15 5 20}{60} frac{70}{60} frac{7}{6}]
Therefore, the average velocity is:
[frac{30}{frac{7}{6}} 30 times frac{6}{7} frac{180}{7} 25 frac{5}{7} text{ km/h}]
Conclusion
Thus, the average velocity of the trip is approximately 25.7143 km/h, or more precisely, 25 5/7 km/h. This detailed calculation method can help in understanding the practical application of velocity and distance in real-life scenarios, making it a valuable lesson for both students and cyclists alike.