Falling Object Height And Average Rate Of Change Formula

Hey guys! Let's dive into a fascinating problem involving a falling object and explore how we can use math to understand its motion. We're given a scenario where an object is dropped from a platform 300 feet above the ground, and we have a function that models its height, h(t), at any given time, t, in seconds. The function is h(t) = 300 - 16t². Our main goal here is to figure out which expression we can use to calculate the average rate at which the object is falling. To really nail this, we'll break down the problem step by step, making sure we understand the physics and the math involved. This isn't just about getting the right answer; it's about grasping the concepts so we can tackle similar problems with confidence. So, let's get started and unravel this problem together!

Understanding the Height Function

Before we jump into calculating the average rate, let's make sure we're all on the same page about what the function h(t) = 300 - 16t² actually tells us. This equation is a classic example of a quadratic function, and in this context, it's modeling the height of a falling object over time. The beauty of mathematics is how it can translate real-world scenarios into concise, understandable formulas. Let's break down each part of this equation so we can see what's really going on.

First, we have the constant term, 300. This represents the initial height of the object when it's dropped. Think of it as the starting point of our object's journey downwards. The object begins its descent from 300 feet above the ground, which is why 300 is our starting value. It's like saying, "Okay, here's our object, floating 300 feet in the air, ready to go!" This initial condition is crucial because it sets the stage for the rest of the object's fall.

Next, we have the term -16t². The '-16' here is actually related to the acceleration due to gravity. On Earth, the acceleration due to gravity is approximately 32 feet per second squared (32 ft/s²). However, in our equation, we see '-16' instead. Why is that? Well, the '-16' comes from taking half of the acceleration due to gravity (32 ft/s²) and including the negative sign because gravity is pulling the object downwards, reducing its height over time. The t² part tells us that the effect of gravity increases quadratically with time. In simpler terms, the longer the object falls, the faster it accelerates downwards. This makes intuitive sense, right? The longer something falls, the more speed it picks up.

So, when we put it all together, h(t) = 300 - 16t² is telling us a story. It says, "Start at 300 feet, and then, as time passes (t seconds), gravity pulls the object down at an increasing rate (-16t²)." This function allows us to plug in any time t and find out the object's height at that exact moment. For example, if we wanted to know the height after 1 second, we'd plug in t = 1: h(1) = 300 - 16(1)² = 300 - 16 = 284 feet. This means after 1 second, the object has fallen from 300 feet to 284 feet.

Understanding this function is key to solving our original problem. We need to know how the height changes over time to calculate the average rate of change. So, with this foundation in place, let's move on to the concept of average rate of change and how it applies to our falling object.

Average Rate of Change: The Key Concept

Alright, now that we've got a handle on our height function, let's talk about what average rate of change actually means. This is a super important concept in calculus and physics, and it's essentially a way of measuring how much a quantity changes, on average, over a specific period. Think of it like this: if you're driving a car, your average speed tells you how far you traveled divided by the time it took. It doesn't tell you your speed at any particular instant, but it gives you an overall sense of your pace.

In mathematical terms, the average rate of change of a function f(x) over an interval [a, b] is defined as the change in the function's value divided by the change in the input value. Sounds a bit formal, right? Let's break it down. The “change in the function’s value” is simply the difference between the function's output at point b and its output at point a, which we write as f(b) - f(a). The “change in the input value” is just the difference between b and a, which is b - a. So, the average rate of change formula looks like this:

Average Rate of Change = (f(b) - f(a)) / (b - a)

This formula is incredibly versatile and can be applied to all sorts of situations, not just math problems. Imagine you're tracking the growth of a plant. You measure its height at the beginning of the week (a) and at the end of the week (b). f(a) would be the height at the beginning, and f(b) would be the height at the end. Plugging these values into our formula gives you the average rate at which the plant grew per day during that week.

Now, let's bring this back to our falling object. In our case, the function is h(t), which represents the height of the object at time t. If we want to find the average rate of change of the object's height between two times, let's say t₁ and t₂, we would use the same formula, but with our specific function:

Average Rate of Change = (h(t₂) - h(t₁)) / (t₂ - t₁)

This expression tells us how much the object's height changed (on average) for each second that passed between t₁ and t₂. It's crucial to remember that this is an average rate. The object's actual speed might vary during the fall due to gravity's increasing effect, but the average rate gives us a good overall picture of the motion.

Think of it like this: if you drove 100 miles in 2 hours, your average speed was 50 miles per hour. But you might have driven faster or slower at different points during the trip. The average just gives you the overall pace.

So, with the concept of average rate of change firmly in our minds, we're ready to apply it to our specific problem. We know the function, and we know the formula. Now, let's see how we can use this to find the expression that represents the average rate at which our object is falling.

Applying the Concept to Our Falling Object Problem

Okay, guys, let's get down to the nitty-gritty and apply what we've learned about average rate of change to our falling object problem. We have the height function, h(t) = 300 - 16t², and we know the general formula for average rate of change: (f(b) - f(a)) / (b - a). Our mission now is to translate this general formula into a specific expression that works for our particular scenario.

Remember, the average rate of change tells us how much the height of the object changes, on average, over a specific time interval. So, we need to pick two points in time, let's call them t₁ and t₂, and calculate the object's height at each of those times. We'll then plug these heights, h(t₁) and h(t₂), into our formula.

Let's start by finding h(t₁). To do this, we simply substitute t₁ into our height function:

h(t₁) = 300 - 16(t₁)²

This gives us the height of the object at time t₁. Now, we do the same thing for t₂:

h(t₂) = 300 - 16(t₂)²

This is the height of the object at time t₂. We've now got the two heights we need to calculate the average rate of change.

Next, we plug these expressions for h(t₁) and h(t₂) into our average rate of change formula:

Average Rate of Change = (h(t₂) - h(t₁)) / (t₂ - t₁)

Substituting our expressions, we get:

Average Rate of Change = ([300 - 16(t₂)²] - [300 - 16(t₁)²]) / (t₂ - t₁)

Now, let's simplify this expression. Notice that we have '300' being added and subtracted in the numerator, so those terms cancel out:

Average Rate of Change = (-16(t₂)² + 16(t₁)²) / (t₂ - t₁)

We can factor out a '-16' from the numerator:

Average Rate of Change = -16[(t₂)² - (t₁)²] / (t₂ - t₁)

Now, we see a difference of squares in the numerator: (t₂)² - (t₁)². We can factor this using the formula a² - b² = (a - b)(a + b):

Average Rate of Change = -16(t₂ - t₁)(t₂ + t₁) / (t₂ - t₁)

Aha! We now have a common factor of (t₂ - t₁) in both the numerator and the denominator, which we can cancel out:

Average Rate of Change = -16(t₂ + t₁)

And there we have it! The expression -16(t₂ + t₁) represents the average rate of change of the falling object's height between times t₁ and t₂. This is the expression we were looking for. It tells us that the average rate of the object's descent depends on the sum of the two times we're considering, multiplied by -16.

So, to recap, we started with the height function, used the definition of average rate of change, plugged in our function, simplified the expression, and arrived at our final answer. This process highlights the power of mathematical formulas to describe and predict real-world phenomena. We've not only solved the problem, but we've also gained a deeper understanding of how the object's motion changes over time.

Conclusion: Putting It All Together

Alright, let's wrap things up and celebrate what we've accomplished! We started with a seemingly complex problem involving a falling object and a height function: h(t) = 300 - 16t². Our goal was to find the expression that represents the average rate at which the object is falling. And guess what? We nailed it!

We began by dissecting the height function, making sure we understood what each part meant. We saw that the '300' represents the initial height, and the '-16t²' term captures the effect of gravity pulling the object downwards. This gave us a solid foundation for understanding the object's motion.

Then, we dove into the concept of average rate of change. We learned that it's a way to measure how much a quantity changes, on average, over a specific interval. We explored the general formula: (f(b) - f(a)) / (b - a) and saw how it applies to various situations, from plant growth to car speeds.

With this knowledge in hand, we applied the average rate of change formula to our falling object problem. We substituted our height function, h(t), into the formula, simplified the expression step-by-step, and ultimately arrived at the answer: -16(t₂ + t₁). This expression tells us the average rate of the object's descent between two times, t₁ and t₂.

But more than just getting the right answer, we've gained a deeper understanding of the physics and math involved. We've seen how a mathematical function can model a real-world phenomenon, and we've learned how to use the concept of average rate of change to analyze the motion of that object. This is the real power of mathematics – it allows us to make sense of the world around us!

So, next time you see an object falling, remember our height function and the average rate of change. You'll have a whole new perspective on what's happening, and you'll be able to explain it using math. That's pretty awesome, right?

Keep exploring, keep questioning, and keep using math to unlock the secrets of the universe. You've got this!