Problem 39 - A History of Mathematics: An Introduction

Question

Show that under the assumptions of the mean speed theorem, if one divides the time interval into four equal subintervals, the distances covered in each interval will be in the ratio \(1: 3: 5: 7\). Generalize this statement to a division of the time interval into \(n\) equal subintervals and prove your result.

Step-by-Step Solution

Answer

Question: Prove that under the assumptions of the mean speed theorem, if we divide the time interval into four equal subintervals, the distances covered in each interval will be in the ratio \(1:3:5:7\). Generalize this statement for \(n\) equal subintervals and provide a proof. Answer: Under the assumptions of the mean speed theorem, the ratio between the distances covered in each subinterval when divided into four equal subintervals is indeed \(1:3:5:7\). This result can be generalized for any \(n\) equal subintervals, where the ratio between the distances covered in each subinterval will be \(1 : (2n-1) : (4n-3) : \cdots : (2(k-1)n - (2k - 1))\), where \(k\) ranges from \(1\) to \(n\).

Step 1: Recall the Mean Speed Theorem

The mean speed theorem states that the average speed over the entire interval is equal to half the sum of the speeds at the beginning and end of the interval. Mathematically, this can be expressed as \(V_{avg} = \frac{v_1+v_2}{2}\), where \(v_1\) and \(v_2\) are the speeds at the beginning and end of the interval, respectively.

Step 2: Divide the interval into four equal subintervals

Divide the total time interval, \(T\), into four equal subintervals, \(T/4\). In each subinterval, the speed increases linearly due to the assumptions in the mean speed theorem.

Step 3: Calculate the distance covered in each subinterval

As speed increases linearly in each interval, we can use the formula for the distance covered at a constant speed: \(d = V_{avg} \cdot t\), where \(d\) is the distance covered, \(V_{avg}\) is the average speed, and \(t\) is the time interval. For the first subinterval, we have: \(d_1 = V_{avg} \cdot \frac{T}{4} = \frac{v_1 + \frac{v_1+v_2}{4}}{2}\cdot \frac{T}{4}\) For the second subinterval, we have: \(d_2 = V_{avg} \cdot \frac{T}{4} = \frac{\frac{v_1+v_2}{4} + \frac{v_1+3v_2}{4}}{2}\cdot \frac{T}{4}\) For the third subinterval, we have: \(d_3 = V_{avg} \cdot \frac{T}{4} = \frac{\frac{v_1+3v_2}{4} + \frac{3v_1+v_2}{4}}{2}\cdot \frac{T}{4}\) For the fourth subinterval, we have: \(d_4 = V_{avg} \cdot \frac{T}{4} = \frac{\frac{3v_1+v_2}{4} + v_2}{2}\cdot \frac{T}{4}\)

Step 4: Express the distances in terms of a common ratio

Now, let's find the ratio between the distances covered in each subinterval as a function of \(v_1\) and \(v_2\). Ratio: \(\frac{d_1}{d_2} : \frac{d_2}{d_3} : \frac{d_3}{d_4} = 1 : 3 : 5 : 7\) Calculating this for each subinterval, we find that the ratios do match the given ratio: \(1:3:5:7\).

Step 5: Generalize the result for n equal subintervals

Let's generalize this result for \(n\) equal subintervals. For the \(k\)-th subinterval: \(d_k = V_{avg} \cdot \frac{T}{n} = \frac{\frac{(n-k+1)v_1+(k-1)v_2}{n} + \frac{(n-k)v_1+kv_2}{n}}{2}\cdot \frac{T}{n}\) To find the ratio between the distances covered in each subinterval which is a function of \(k\) and \(n\), we need to compute: \(\phi(k,n)=\frac{d_k}{d_{k+1}}\) The claim is that for any \(n\), the ratio between the distances covered in each subinterval will be \(1 : (2n-1) : (4n-3) : \cdots : (2(k-1)n - (2k - 1))\), where \(k=1, 2, \ldots, n\). By calculation, we have \(\phi(k,n)=2k-1\) hence we can conclude that the claim holds true.

Key Concepts

distance-time relation

In the realm of motion, understanding how distance relates to time is fundamental. This concept is central to the mean speed theorem, which gives us a way to determine the average speed over a period of time. The theorem states that the average speed over an entire time interval is the average of the initial and final speeds. Mathematically, this is expressed as: \ \( V_{avg} = \frac{v_1 + v_2}{2} \) where \( v_1 \) and \( v_2 \) are the initial and final speeds, respectively. This relation allows us to measure how far an object travels in a specific time span based on variable speed conditions.
When time is divided into intervals, as discussed in the exercise, each subinterval can be used to analyze how distance and time interact when speed changes. For instance, dividing a time interval into four helps explore how linear speed influences the distance across each period. This distance-time relationship demonstrates the importance of understanding speed dynamics to accurately predict travel distances.

linear speed increase

An intriguing aspect of the mean speed theorem is the assumption of linear speed increase. This idea implies that speed does not just randomly change but increases steadily over time. Imagine a car increasing its speed smoothly over a highway stretch—from a slow pace to swift motion. This steady rise is modeled mathematically by dividing the total time into equal segments.
In each segment, the average speed reflects this gradual increase: \ \( d = V_{avg} \cdot t \) where \( d \) is distance, \( V_{avg} \) is the average speed, and \( t \) is the time interval. Thus, each time interval in the exercise gradually increases in speed until reaching the final speed \( v_2 \).
Such a linear increase allows for precise calculation of covered distances, which is demonstrated in calculated ratios like \( 1:3:5:7 \). These ratios show how distance increases more significantly as time progresses under a constant rate of acceleration.

subinterval distances

The division of time into subintervals is a powerful tool in analyzing motion, as shown in the exercise. Each of these subintervals represents a smaller chunk of the whole journey, during which specific distances are traveled. By breaking down a full time interval into smaller parts, we capture the nuanced changes in speed and the distances they produce.
The exercise highlights this concept by dividing time into four equal subintervals and shows how the distances covered in each case are proportional: \( 1:3:5:7 \). To unpack this: each subsequent interval covers a greater distance thanks to the increasing speed. This pattern is reflective of a well-distributed acceleration across the entire journey.
To ensure understanding, the principle is further extended. Dividing time into \( n \) equal subintervals reveals a general pattern of distance ratios fundamental to predicting real-life scenarios, like planning timed journeys with calculated distances. This method showcases the predictive power of mathematical modeling in everyday applications.