Problem 41 - A History of Mathematics: An Introduction

Question

Solve the following problem of Oresme: Divide the line \(A B\) of length 1 (representing time) proportionally to infinity in a ratio of \(2: 1\); that is, divide it so the first part is one-half, the second one-quarter, the third one-eighth, and so on. Let there be a given finite velocity (say, 1 ) in the first interval, a uniformly accelerated velocity (from 1 to 2 ) in the second, a constant velocity (2) in the third, a uniformly accelerated velocity (from 2 to 4 ) in the fourth, and so on (Fig. 10.18). Show that the total distance traveled is \(7 / 4\).

Step-by-Step Solution

Answer

Answer: The total distance traveled is 7/4.

Step 1: Setup the series for the distance traveled in each interval

For each partition of the line segment, the length of the interval can be calculated using the following series: 1/2, 1/4, 1/8, ... The velocity in each interval is given by the problem statement. Now we need to find the distance traveled in each interval, which is given by the product of its length and the corresponding velocity. We can represent this as a series: \(\text{Distance} = \sum_{n=1}^{\infty} \text{Velocity} \times \text{Length}\)

Step 2: Calculate the distance traveled in each interval

Using the information from the problem statement, let's calculate the distance traveled for each interval: 1. Interval 1 (Length: 1/2, Velocity: 1): Distance = 1 * 1/2 = 1/2 2. Interval 2 (Length: 1/4, Accelerated Velocity: 1→2): Distance = (1/4) * (1 + 2) / 2 = 3/8 3. Interval 3 (Length: 1/8, Velocity: 2): Distance = 2 * 1/8 = 1/4 4. Interval 4 (Length: 1/16, Accelerated Velocity: 2→4): Distance = (1/16) * (2 + 4) / 2 = 3/16 Since the sequence repeats itself from now on (i.e., constant velocity, then accelerated velocity), we can group them in two and write the series as: \(\text{Distance} = 1/2 + 3/8 + \frac{1}{4}\sum_{n=0}^{\infty} \left(\frac{1}{2^n}\right)\)

Step 3: Evaluate the infinite series and find the total distance traveled

We are now left to evaluate the infinite geometric series of the sequence, i.e., \(\sum_{n=0}^{\infty} \left(\frac{1}{2^n}\right)\). This series converges since it is a geometric series with a common ratio less than 1 (r = 1/2). The sum of the infinite geometric series can be calculated using the formula: \(\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}\), where a is the first term and r is the common ratio. For our series, a = 1 and r = 1/2. So we can calculate the sum of the series as: \(\sum_{n=0}^{\infty} \left(\frac{1}{2^n}\right) = \frac{1}{1 - 1/2} = 2\) Now, calculating the total distance traveled: \(\text{Total Distance} = 1/2 + 3/8 + \frac{1}{4}(2) = 1/2 + 3/8 + 1/2 = 7/4\) Therefore, the total distance traveled is 7/4.

Key Concepts

Geometric Series

A geometric series is a sum of terms where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the context of this problem, the lengths of the line segments to be divided (1/2, 1/4, 1/8,...) form a geometric series, which makes the infinite series manageable to calculate. By summing the lengths:

• First term = 1/2
• Common ratio = 1/2

We use the formula to calculate the sum of an infinite geometric series:\[ S = \frac{a}{1 - r} \]where \(a\) is the first term and \(r\) is the common ratio. Here, you'll see how these segments effectively sum to 1, representing the total time line over which all movements are calculated.

Velocity

Velocity is a measure of how fast something is moving in a specified direction. In this problem, different velocities are assigned to different segments of time. For a geometric series of time intervals, understanding the concept of velocity is crucial:

• Interval 1: Constant velocity of 1 unit.
• Interval 2: Velocity accelerates uniformly from 1 to 2.
• Interval 3: Constant velocity of 2 units.
• Interval 4: Velocity accelerates uniformly from 2 to 4.

These variances in velocity affect how much distance is traversed during each interval. The essential takeaway here is that velocity isn't just a number; it's a vital part of understanding the movement over time and directly relates to the distances covered in this mathematical puzzle.

Distance Calculation

When calculating distance in physics, the formula often used is \( \text{Distance} = \text{Velocity} \times \text{Time} \). In this exercise, time is represented by each segment's length from the geometric series, and various velocities for each segment:

• Interval 1: Distance = 1/2\( \times \)1 = 1/2
• Interval 2: Distance = 1/4\(\times\)Average Velocity = 3/8
• Interval 3: Distance = 1/8\(\times\)2 = 1/4

Distance for each segment is derived from its length and velocity, whether constant or accelerating. This approach connects the mathematical series with real-world physics, showing how different segments add up to achieve the final total distance of \(7/4\), providing a comprehensive answer to the problem.

Proportional Division

Proportional division involves partitioning something in proportion to specified ratios. In this problem, you are tasked with dividing a line, representing time, into an infinite number of segments:

• First segment takes up 1/2 of the total distance.
• Second segment is 1/4, and third is 1/8, continuing in this manner.

Proportional division helps maintain structured distribution across what would otherwise be infinitely divided segments. By maintaining these proportions, you can easily calculate distance and connect them with velocities, facilitating the understanding of both mathematical and physical concepts in distance traveled over time, as showcased in this specific exercise. This understanding simplifies complex problems into more manageable calculations.