Calculating the Average Speed of a Body Traveling Variable Distances
In physics, understanding the behavior of motion over different terrains and speeds is a fundamental concept. This article explores how to calculate the average speed of a body that travels the first half of a distance at a different speed than the second half. We will apply this knowledge using a practical example to understand the theoretical outcome.
Problem Statement: Variable Speed Over Half Distances
A body covers the first half of the distance between two places at 4 m/s and the second half at 60 m/s. What is the average speed of the body during the journey?
Step-by-Step Solution
Let us assume the total distance traveled by the body be (x).
Case I: First Half of the Distance
Distance ((S')) (x/2)
Velocity ((v)) 4 m/s
Time ((t)) (frac{distance}{velocity} frac{x/2}{4})
Time (frac{x}{8}) seconds
Case II: Second Half of the Distance
Distance ((S')) (x/2)
Velocity ((v)) 60 m/s
Time ((t')) (frac{distance}{velocity} frac{x/2}{60})
Time (frac{x}{12}) seconds
Total Time Taken to Travel
Total time ((T)) (t t' frac{x}{8} frac{x}{12})
Total time (x(frac{1}{8} frac{1}{12}))
Total time (x(frac{3}{24} frac{2}{24}) frac{5x}{24}) seconds
Calculating the Average Speed
Average speed (frac{total distance}{total time taken})
Average speed (frac{x}{frac{5x}{24}})
Average speed (frac{24x}{5x} frac{24}{5} 4.8) m/s
Conclusion
The average speed of the body during the journey is 4.8 m/s. This problem demonstrates the fundamental principle that varying speeds over the same distance result in a unique average speed that cannot be simply averaged and is essential for a deeper understanding of motion concepts.