Problem 15 - Applied Calculus

Question

Write a formula representing the function. The average velocity, \(v\), for a trip over a fixed distance, \(d\), is inversely proportional to the time of travel, \(t .\)

Step-by-Step Solution

Answer

The function is given by the formula: \(v = \frac{d}{t}\).

Step 1: Understand Proportional Relationship

The problem states that average velocity, \(v\), is **inversely proportional** to the time of travel, \(t\). This means that as the time increases, the average velocity decreases and vice versa. Mathematically, when a value is inversely proportional to another, the product of the two values is a constant.

Step 2: Express the Inverse Proportionality Mathematically

To express that \(v\) is inversely proportional to \(t\), we use the formula \(v = \frac{k}{t}\), where \(k\) is the proportionality constant. This equation shows that the product \(v \times t = k\) remains the same regardless of the individual values of \(v\) or \(t\).

Step 3: Relate Constant to Fixed Distance

Since \(k\) is a constant representing the product \(v \times t\), and the average velocity \(v\) is defined by the formula \(v = \frac{d}{t}\), where \(d\) is a fixed distance, we equate the two expressions for \(v\): \[ \frac{k}{t} = \frac{d}{t} \].

Step 4: Establish the Connection between Proportionality Constant and Distance

By equating \(\frac{k}{t} = \frac{d}{t}\), we realize that \(k\) must be equal to \(d\), given that \(t\) cancels out on both sides. Therefore, the constant \(k\) represents the fixed distance \(d\), making the formula \(v = \frac{d}{t}\), which is consistent with the definition of average velocity.

Key Concepts

Average Velocity

Average velocity is an essential concept in studying motion, as it provides insight into how fast an object or traveler moves over a specific period of time. Average velocity is defined as the total distance traveled divided by the total time taken to travel that distance. Since distance, in this problem, is a fixed value, the average velocity can only change if the time taken changes.
In a mathematical sense, if you want to calculate the average velocity, you use the formula:

• \( v = \frac{d}{t} \)

Where \( v \) is the average velocity, \( d \) is the fixed distance, and \( t \) is the time taken.

From this, you can see that average velocity is inversely proportional to time, meaning if you increase the time, the velocity decreases and vice versa. This inverse relationship is crucial in understanding movement over a set distance.

Proportionality Constant

The proportionality constant in the context of motion and average velocity is pivotal. Inverse proportionality is when two quantities multiply together to always give the same product. In this case, when average velocity \( v \) is inversely proportional to time \( t \), their product is a constant value:

• \( v \times t = k \)

Here, \( k \) represents the proportionality constant. It remains constant regardless of specific values of velocity or time, reflecting an inherent property of motion over the fixed distance.

By rearranging the inverse proportionality formula to express average velocity, we see:

• \( v = \frac{k}{t} \)

This highlights how any change in time will inversely affect velocity, maintaining the same proportionality constant defined by the fixed distance. The constant \( k \) bridges the relationship between variables, facilitating our understanding of motion equilibrium.

Fixed Distance

The notion of fixed distance plays a central role in analyzing average velocity across travel. A fixed distance means that the length of the path an object or traveler covers does not change. Hence, it becomes vital to understand the interaction between distance and other factors like time and velocity.

When we deal with a fixed distance in our equation, it solidifies our interpretations of inverse proportionality in motion. Fixing distance simplifies the average velocity formula:

• \( v = \frac{d}{t} \)

Given that the distance \( d \) is constant, the average velocity is affected by any variations in time taken over that fixed distance.

Moreover, by equating the expressions for average velocity in inverse proportionality:

• \( \frac{k}{t} = \frac{d}{t} \),

it becomes evident that the proportionality constant \( k \) is equivalent to the fixed distance \( d \). This simplification of concepts helps in solid planning and analysis of speeds and times for any journey covering a specific path.