A Thought-Provoking Problem: Calculating Average Speed over Infinite Equal Parts

This problem presents a fascinating challenge in the realm of calculus and physics, inviting us to explore average speed across an intriguing scenario where a trip is divided into infinitely small parts and each part is traveled at a gradually decreasing speed.

Let's first consider the initial simplification where the speed decreases linearly over time. If a person divides a trip into infinite equal parts and travels the first part at 50 kph and the last part at 30 kph, how do we calculate the average speed of the entire trip?

Constant Rate of Decrease - A Simple but Insightful Approach

The initial approach might lead us to assume a constant rate of decrease in speed. In such a scenario, the speed decreases linearly from 50 km/h to 30 km/h over the duration of the trip, which can be mathematically represented. The average speed in this case can be easily calculated using the formula for the average value of a function, which is the 'area under the curve' divided by the interval. For a simple linear decrease, the average speed is simply the average of the initial and final speeds.

average speed (v0 v1) / 2 (50 30) / 2 40 kph

However, visualizing the problem in terms of a graph where time is on the x-axis and speed is on the y-axis, we can see the triangular area formed with the x-axis, leading to the same result.

Decreasing Speed with Distance - A More Complex Scenario

Now, if the speed decreases smoothly with distance rather than time, the problem requires a more rigorous mathematical treatment. Here, the initial conditions remain the same: the trip begins at 50 km/h and ends at 30 km/h. But the speed varies smoothly with distance.

To tackle this, we need to derive the velocity as a function of time. Given that the trip distance is (L), and the speed reduces from 50 km/h at the start to 30 km/h at the end, we can express the rate of change of velocity with distance as a constant (k):

We have: v0 50 km/h, v1 30 km/h, x0 0, x1 L, t0 0, t1 unknown.

The rate of change of velocity with distance is given by:

dv/dx (v1 - v0) / L k

The differential equation resulting from this expression can be solved to find the velocity as a function of time. Using the initial and final conditions, we can integrate and find the velocity function:

vt v0e^(kt)

Substituting (t1) and solving for (t1), we find the total time taken for the trip:

t1 (1/k)ln(v1/v0)

The average speed can then be calculated as the total distance divided by the total time, which simplifies to:

vavg (v1 - v0) / (1/k)ln(v1/v0) (30 - 50) / (1/k)ln(30/50) 39.1523 kph

Conclusion

Despite the complexity, the average speed does not depend on the length of the trip, but rather on the initial and final speeds. This problem highlights how calculus can be used to solve seemingly complex problems in an elegant manner, emphasizing the power of mathematical modeling and problem-solving techniques.

Thanks for sharing this problem! It's not only an opportunity to exercise our mathematical muscles but also to appreciate the beauty of physics and calculus in real-world scenarios.