Ball Drop: Finding Time At 50 Meters

Hey Plastik Magazine readers! Let's dive into a fun little physics problem involving a falling ball. We're given a quadratic model that describes the height of the ball after it's dropped, and our mission is to figure out when it reaches a specific height. Ready? Let's get started!

Understanding the Quadratic Model

Okay, so we have this equation: f(x) = -5x² + 200. What does it all mean? Well, in this scenario, f(x) represents the height of the ball in meters, and x is the time in seconds after the ball is dropped. The equation itself is a quadratic function, which means the graph of the height over time will be a parabola. The negative coefficient in front of the x² term tells us that the parabola opens downwards, which makes sense since the ball's height decreases as time goes on.

Now, let's break down the components. The -5x² part indicates that the height decreases proportionally to the square of the time. This is due to gravity accelerating the ball downwards. The +200 is a constant term, and it represents the initial height of the ball when it's first dropped (at x = 0 seconds). So, initially, the ball is 200 meters above the ground. This model f(x) = -5x² + 200 is a simplified representation, neglecting air resistance and other factors, but it’s good enough for our theoretical exploration.

Think of it like this: at the very beginning (x = 0), the height is 200 meters. As time increases, the x² term grows, and since it's multiplied by -5, the overall value decreases, reducing f(x). This decrease represents the ball falling closer to the ground. The question we're tackling is finding the specific x value (time) when f(x) equals 50 meters. Why 50 meters? It's just an arbitrary height we've chosen to investigate, but it helps us understand how to work with the model and extract meaningful information. Understanding this model allows us to predict the height of the ball at any given time, assuming the conditions remain consistent.

Setting Up the Equation

The problem states that we want to find the time (x) when the ball is 50 meters from the ground. In mathematical terms, we want to find x when f(x) = 50. So, we can set up the equation like this:

50 = -5x² + 200

This equation tells us that we're looking for the value of x that makes the expression -5x² + 200 equal to 50. Now, our goal is to isolate x and solve for it. We'll do this by rearranging the equation and using some basic algebra.

Firstly, let's subtract 200 from both sides of the equation to get the x² term by itself:

50 - 200 = -5x²

This simplifies to:

-150 = -5x²

Next, we'll divide both sides by -5 to get x² alone:

-150 / -5 = x²

Which simplifies to:

30 = x²

Now we have a simple equation: x² = 30. To solve for x, we need to take the square root of both sides. Remember that when we take the square root, we usually consider both positive and negative solutions. However, in this context, since x represents time, we can disregard the negative solution because time cannot be negative.

Solving for Time (x)

Alright, guys, where were we? Oh yeah, solving for x. So, we've got x² = 30. To find x, we need to take the square root of both sides. Remember, when taking the square root, we generally consider both positive and negative solutions. However, in this real-world scenario, x represents time, and time can't be negative (at least not in this context!). So, we'll only focus on the positive square root.

Taking the square root of both sides gives us:

x = √30

Now, let's calculate the square root of 30. You can use a calculator for this, or if you're feeling old-school, you can estimate it. The square root of 30 is approximately 5.477.

So, we have:

x ≈ 5.477

Since the question asks for an approximate answer, we can round this to two decimal places, giving us:

x ≈ 5.48

Therefore, the ball is approximately 50 meters from the ground after about 5.48 seconds.

Choosing the Correct Answer

Okay, now let's look at the answer choices provided:

A. 2.45 B. 3.16 C. 5.48 D. 7.07

Comparing our calculated value of approximately 5.48 seconds to the answer choices, we see that option C, 5.48, matches our result. Therefore, the correct answer is C. It's always a good idea to double-check your work and make sure the answer makes sense in the context of the problem. In this case, 5.48 seconds seems reasonable for a ball falling from an initial height of 200 meters to a height of 50 meters. Remember, the other options are there to throw you off, so stay focused and trust your calculations.

Conclusion

So, there you have it! The ball is approximately 50 meters from the ground after about 5.48 seconds. We solved this problem by understanding the quadratic model, setting up the equation correctly, and using basic algebra to isolate the variable we wanted to find. I hope you found this breakdown helpful and insightful. Remember, practice makes perfect, so keep honing those problem-solving skills!