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 formula is \( v = \frac{d}{t} \).

Step 1: Define the relationship

The problem states that the average velocity \( v \) is inversely proportional to the time of travel \( t \). This means that as \( t \) increases, \( v \) decreases, and vice versa. Mathematically, this relationship is expressed as \( v \propto \frac{1}{t} \).

Step 2: Introduce the proportionality constant

To convert the proportionality into an equation, we introduce a constant of proportionality, \( k \). This leads to the equation: \( v = \frac{k}{t} \).

Step 3: Determine the constant using the fixed distance

Since the average velocity is for a trip over a fixed distance \( d \), we can relate this to the formula \( v = \frac{d}{t} \), a standard equation for velocity. Thus, \( k = d \).

Step 4: Write the final formula

Substitute \( k = d \) into the equation \( v = \frac{k}{t} \) to get the final relationship for average velocity over a fixed distance: \( v = \frac{d}{t} \).

Key Concepts

Inverse Proportionality

Imagine two quantities that always multiply to give the same number, such as a constant value. This is what inverse proportionality is all about. When average velocity \( v \) is inversely proportional to time \( t \), it implies a special relationship between them:

• If you increase the time \( t \), the average velocity \( v \) must decrease to keep the product \( v \cdot t \) constant.
• Conversely, if you decrease the time \( t \), then \( v \) should increase to maintain the same constant product \( v \cdot t \).

This relationship is mathematically captured as \( v \propto \frac{1}{t} \). The key point to remember is that an increase in one results in a proportional decrease in the other, and this type of relationship is vital in controlling and predicting the behavior of systems where inverse proportionality occurs.

Distance-Time Relationship

The distance-time relationship is foundational in understanding motion and velocity. It's rooted in the basic formula for velocity: \( v = \frac{d}{t} \), where \( v \) stands for velocity, \( d \) stands for distance, and \( t \) stands for time. This equation reveals how distance traveled in motion is directly related to both velocity and time.

To comprehend this concept further, consider these points:

• For a fixed distance \( d \), if time \( t \) is reduced, velocity \( v \) must increase to cover the same distance.
• A longer time \( t \) to cover the same distance \( d \) results in a decreased velocity \( v \).

These observations highlight why the distance-time relationship is crucial to understanding how objects move in the real world. Changes in any of the three values directly affect the others, allowing us to predict how fast or slow something is moving.

Velocity Equation

The velocity equation, \( v = \frac{d}{t} \), emerges from combining the insights about proportionality and distance-time relationships. It succinctly captures how velocity is influenced by both distance and time. Here's a breakdown:

• \( d \) is the distance: the length over which the object moves.
• \( t \) is the time: the duration over which the movement occurs.
• \( v \) is the velocity: how quickly the object travels over distance \( d \) during time \( t \).

This handy equation is crucial for solving problems where knowing how fast something moves (velocity) is essential given the distance and time. By understanding this formula, you gain insight into the basic physics of movement and can calculate how changes in distance or time affect the overall speed of travel. The beauty of this equation is its simplicity and clarity, making it an indispensable tool in physics and everyday applications alike.