Introduction
Let's dive into a classic problem involving harmonic motion, specifically modeling the vertical movement of a buoy in the sea. This is a fantastic way to apply our understanding of trigonometric functions like sine and cosine to real-world scenarios. We're given that a buoy starts at sea level (height 0), oscillates vertically with a maximum displacement of 8 feet, and takes 4 seconds to travel from its highest to its lowest point. Our goal is to determine the equation that accurately describes this motion. Guys, let's break it down step by step so it's super clear!
Understanding the Buoy's Motion
First, visualize the buoy bobbing up and down in the water. The fact that it starts at a height of 0 and then moves upwards suggests we're dealing with a sinusoidal function. Harmonic motion is characterized by smooth, repetitive oscillations, and sine and cosine functions are perfect for modeling this. The buoy's maximum displacement of 8 feet tells us the amplitude of the motion – the distance from the equilibrium position (sea level) to the highest or lowest point. This amplitude is crucial because it directly scales the trigonometric function. Think of it as how far the wave stretches vertically. Moreover, the time it takes for the buoy to move from its highest to its lowest point is directly related to the period of the oscillation. This is where things get interesting because we're given half the period (4 seconds), not the full period. We need to remember that the full period is the time it takes for one complete cycle (from highest to lowest and back to highest).
The amplitude is basically the maximum displacement from the center line, which in our case is 8 feet. That’s how high or low the buoy goes from its starting point. The information about the time to go from the highest to the lowest point is super important for figuring out the period. Remember, the period is the time for one complete cycle – up and down. Since it takes 4 seconds to go from the highest to the lowest, the full cycle (highest to lowest and back to highest) would take twice as long. That means the period is 8 seconds. This is essential for determining the coefficient of our variable (often 't' for time) inside the trigonometric function. Now, let's consider the general form of the sinusoidal equation we'll use to model this. We know it's going to involve either sine or cosine, and it'll look something like this: y = A * sin(B * t) or y = A * cos(B * t), where A is the amplitude, B is related to the period, and t is time. We need to figure out the values of A and B based on the information we have. So, the amplitude (A) is straightforward – it's 8 feet. The value of B is a bit trickier but can be found using the relationship between the period (T) and B: B = 2π / T. We've already figured out that the period (T) is 8 seconds, so we can plug that into the formula to find B.
Determining the Correct Equation
Now, let’s convert our insights into a mathematical equation. Since the buoy starts at a height of 0 and initially moves upwards, a sine function is a natural fit. The general form of the equation we're looking for is y = A * sin(Bt), where:
- y is the height of the buoy at time t
- A is the amplitude (maximum displacement)
- B is related to the period of the motion
- t is the time in seconds
We know the amplitude A is 8 feet. To find B, we use the relationship between the period (T) and B: B = 2π / T. The period T is the time it takes for one complete cycle, which is twice the time it takes to go from the highest to the lowest point. So, T = 2 * 4 seconds = 8 seconds. Plugging this into the formula for B, we get B = 2π / 8 = π / 4. Therefore, the equation that models the buoy's motion is y = 8 * sin((π / 4) * t). This equation tells us the height of the buoy at any given time t. We can plug in different values of t to see how the buoy moves up and down over time. For instance, at t = 0, y = 8 * sin(0) = 0, which matches our initial condition that the buoy starts at sea level. At t = 2, which is a quarter of the period, y = 8 * sin(π / 2) = 8, meaning the buoy is at its highest point. At t = 4, which is half the period, y = 8 * sin(π) = 0, bringing the buoy back to sea level. At t = 6, which is three-quarters of the period, y = 8 * sin(3π / 2) = -8, putting the buoy at its lowest point. These values confirm that our equation accurately models the buoy's motion.
Analyzing Potential Equation Choices
Often, problems like these present multiple equation choices, and we need to strategically eliminate the incorrect ones. The amplitude is a great starting point. Any equation that doesn't have 8 as the coefficient of the sine or cosine term is immediately out. Next, we focus on the period. We calculated B as π / 4, which means the term inside the sine function should be (π / 4) * t. Any equation with a different coefficient for t is incorrect. It’s a process of elimination that makes the problem much more manageable. For example, if we saw an equation with a coefficient of 4 instead of 8 for the amplitude, we’d know it’s wrong right away. Similarly, if the term inside the sine function was (π / 2) * t, that would indicate a different period, and we’d know that equation is incorrect as well. Guys, breaking it down like this, we can confidently choose the right answer even if we're faced with tricky options.
In summary, the key is to understand the physical meaning of the parameters in the sinusoidal equation. The amplitude determines the vertical stretch, the period dictates the frequency of oscillations, and the sine or cosine function determines the starting point of the motion. By carefully analyzing the problem statement and using these principles, we can confidently model harmonic motion and solve related problems.
Conclusion
Modeling the buoy's motion demonstrates a practical application of trigonometric functions. By understanding the concepts of amplitude, period, and phase shift, we can accurately describe oscillatory behavior in various real-world scenarios. This problem highlights the power of mathematics in representing and predicting physical phenomena. So, guys, next time you see a buoy bobbing in the water, remember the math behind its motion!
Keywords Discussion
Original Keyword: Which of the following equations can
Rewritten Keyword: Which equation models the buoy's motion?
This rewritten keyword is clearer and more specific, directly asking for the equation that models the buoy's movement. It avoids the vagueness of the original and aligns better with the problem's objective. Using keywords like "models" and "motion" helps focus the query on the relevant concepts.