Inverse Relationship: Solving For The Constant With Gamma Rays

Hey guys! Let's dive into an interesting physics problem today. We're going to explore the inverse relationship between wavelength and frequency, specifically focusing on gamma rays. If you've ever wondered how these two properties are connected, or how to calculate the constant that ties them together, you're in the right place. We'll break it down step by step, so it's super easy to follow. So, grab your thinking caps, and let's get started!

Understanding Inverse Relationships in Physics

In physics, an inverse relationship means that when one quantity increases, the other decreases, and vice versa. Think of it like a seesaw – when one side goes up, the other goes down. In the context of waves, particularly electromagnetic radiation like gamma rays, this relationship exists between wavelength and frequency.

Wavelength, often represented by the Greek letter lambda (λ), is the distance between two consecutive crests or troughs of a wave. It's essentially the length of one complete wave cycle. On the other hand, frequency, usually denoted by the letter f, is the number of waves that pass a given point per unit of time, typically measured in Hertz (Hz), which represents cycles per second.

The inverse relationship between these two is fundamental. Imagine a wave being stretched out – its wavelength increases. If the speed of the wave remains constant, then fewer wave cycles will pass a point per second, meaning the frequency decreases. Conversely, if the wave is compressed, its wavelength decreases, and more cycles pass per second, increasing the frequency. This inverse proportionality is a crucial concept in understanding wave behavior and is mathematically expressed through a constant.

To really grasp this, think about different types of electromagnetic radiation. Radio waves, for example, have long wavelengths and low frequencies. On the opposite end of the spectrum, gamma rays have extremely short wavelengths and very high frequencies. This vast difference in wavelength and frequency underscores the inverse nature of their relationship. We see this relationship in everyday applications, from tuning a radio (adjusting frequency to receive different stations) to medical imaging (using the properties of different wavelengths for diagnosis).

Setting Up the Problem: Wavelength, Frequency, and the Constant

Before we jump into solving the problem, let's clearly define what we're working with. We're given that wavelength (x) and frequency (y) are inversely related. This means their relationship can be expressed by the equation:

y = k / x

Where:

• y represents the frequency.
• x represents the wavelength.
• k is the constant of proportionality. This is the value we're trying to find. It essentially quantifies the specific relationship between frequency and wavelength for the type of radiation we're dealing with, which in this case, is gamma rays.

We're also given specific values for frequency and wavelength for a particular gamma ray: Frequency (y) = 40 Hz Wavelength (x) = 300,000 The key here is understanding that the constant k is unique to the medium and the type of wave. For electromagnetic waves like gamma rays traveling through a vacuum (or air, to a close approximation), this constant is related to the speed of light. However, the problem is designed to have us calculate k directly from the given values, reinforcing the concept of inverse proportionality.

Our goal is to use the provided values of frequency and wavelength to solve for k. We'll do this by rearranging the equation and plugging in the known values. This process will not only give us the numerical value of k but also solidify our understanding of how the inverse relationship works in practice. By finding k, we essentially determine the factor that connects the frequency and wavelength of gamma rays under the given conditions. So, let's get to the calculation and see what we get!

Solving for the Constant (k) Step-by-Step

Alright, let's get our hands dirty with the math! We know that the relationship between frequency (y) and wavelength (x) is given by:

y = k / x

Our goal is to find the value of k, the constant of proportionality. To do this, we need to rearrange the equation so that k is isolated on one side. We can achieve this by multiplying both sides of the equation by x:

y * x = (k / x) * x

This simplifies to:

k = y * x

Now we have an equation that directly expresses k in terms of y and x. This is fantastic because we're given the values for both frequency (y) and wavelength (x). Remember, we have:

• Frequency (y) = 40 Hz
• Wavelength (x) = 300,000

Next, we simply substitute these values into our equation:

k = 40 * 300,000

Now it's just a matter of performing the multiplication:

k = 12,000,000

So, there you have it! The constant of proportionality, k, for this particular gamma ray is 12,000,000. This value represents the specific relationship between frequency and wavelength for these gamma rays. It essentially tells us that the product of the frequency and wavelength will always be this value, reflecting the inverse relationship we discussed earlier. Isn't it cool how we can use a simple equation to quantify such a fundamental property of waves?

Choosing the Correct Answer and Implications

Now that we've calculated the value of k, let's go back to the multiple-choice options given in the problem:

A. k = 0.0001 B. k = 7,500 C. k = 12,000,000 D. k = 40,000,000

Based on our calculation, the correct answer is C. k = 12,000,000. It's always a good feeling when your calculated answer matches one of the options, right? This confirms that we've applied the inverse relationship formula correctly and performed the calculations accurately.

But beyond just getting the right answer, it's important to understand what this result implies. The constant k in this context is directly related to the speed of the wave. In the case of electromagnetic radiation like gamma rays, this speed is the speed of light in the given medium (which is approximately the speed of light in a vacuum in this case). While the problem doesn't explicitly state it, this constant is numerically equal to the speed of light (approximately 3 x 10^8 meters per second) when wavelength is measured in meters and frequency in Hertz.

The large value of k reflects the incredibly high speed at which gamma rays travel. This understanding is crucial in various applications, from medical imaging and cancer treatment to understanding astrophysical phenomena. The inverse relationship between wavelength and frequency, governed by this constant, dictates how these rays interact with matter and how we can utilize them in different technologies. So, by solving this problem, we've not only practiced our algebra skills but also gained a deeper appreciation for the fundamental physics that governs the behavior of electromagnetic radiation. Great job, guys!