Finding The Max Height Of A Kicked Football

Hey guys! Ever wondered how high a football can go when kicked straight up? Let's dive into the math behind this and figure out the maximum height the ball reaches. We're given a scenario where a football is kicked upwards with an initial velocity, and we'll use a handy formula to find the peak of its flight. This is a classic physics problem, and it's super cool to see how math helps us understand the real world!

Understanding the Problem: The Basics of Vertical Motion

Okay, so the problem tells us a football is kicked upwards with an initial speed of 96 feet per second (ft/sec) from a height of 6 feet. The height of the ball above the earth is described by the equation h(t) = -16t² + 96t + 6. Here, 'h(t)' represents the height at any given time 't' (in seconds). The equation is a quadratic equation, which means it describes a parabola. In this case, the parabola opens downwards (because of the negative sign in front of the t² term), and its vertex represents the maximum height the ball reaches. This is where our journey to finding the maximum height begins!

This type of problem falls under the category of projectile motion, a core concept in physics. Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity. In our scenario, the football is the projectile. The initial velocity of the kick, the force of gravity, and the initial height all play crucial roles in determining the ball's trajectory and, consequently, its maximum height. The equation provided, h(t) = -16t² + 96t + 6, is a mathematical model of this physical phenomenon. The -16t² term represents the effect of gravity, the 96t term accounts for the initial upward velocity, and the +6 is the initial height. Understanding these components gives us the foundation needed to solve the problem and truly comprehend how the ball moves through the air.

To find the maximum height, we're essentially looking for the vertex of this parabola. There are several ways to do this, and we'll explore the most straightforward method. We can use the vertex formula, complete the square, or use calculus to find the time at which the ball reaches its maximum height, which allows us to determine the maximum height. Let's get started with finding the vertex because, you know, it's what we are after! Remember, understanding the problem is as important as solving it. By breaking down the problem into its components and understanding the physics at play, we can approach the solution with confidence and clarity. So, let's gear up and start our ascent to finding the maximum height of the football!

Methods to Determine Maximum Height

There are a couple of ways we can find the maximum height. Each method uses different mathematical principles, but all lead to the same answer. It's like taking different paths up a mountain; you still reach the peak. So, let's look at the methods.

Method 1: Using the Vertex Formula

The vertex formula is a handy tool specifically designed for quadratic equations like ours. The vertex formula for a parabola in the form of h(t) = at² + bt + c is given by t = -b / 2a. This gives us the time ('t') at which the maximum height occurs. Once we have that time, we can plug it back into the original equation to find the maximum height. For our equation, h(t) = -16t² + 96t + 6, 'a' is -16 and 'b' is 96.

Let's calculate the time first: t = -b / 2a = -96 / (2 * -16) = -96 / -32 = 3 seconds.

This tells us that the ball reaches its maximum height at 3 seconds. Now, we plug this value of 't' back into the original equation to find the maximum height:

h(3) = -16(3)² + 96(3) + 6 h(3) = -16(9) + 288 + 6 h(3) = -144 + 288 + 6 h(3) = 150 feet.

So, according to this method, the maximum height reached by the ball is 150 feet. Easy peasy, right? The vertex formula is a direct and efficient way to solve this type of problem. It directly gives us the time at which the maximum height is achieved, and then we just need to substitute that time back into the equation. It's like having a shortcut in a maze; you quickly get to the end! So let's compare with other methods.

Method 2: Completing the Square

Completing the square is another approach to finding the vertex of a parabola. It involves rewriting the quadratic equation into vertex form, which directly reveals the coordinates of the vertex. While this method might seem a bit more involved, it provides a deeper understanding of the quadratic equation's structure and behavior.

Here’s how we do it with our equation h(t) = -16t² + 96t + 6.

First, factor out the coefficient of t² from the first two terms: h(t) = -16(t² - 6t) + 6.

Next, complete the square inside the parentheses. Take half of the coefficient of 't' (-6), square it ((-3)² = 9), and add and subtract it inside the parentheses: h(t) = -16(t² - 6t + 9 - 9) + 6.

Now, rewrite the perfect square trinomial as a squared term and simplify: h(t) = -16((t - 3)² - 9) + 6.

Distribute the -16: h(t) = -16(t - 3)² + 144 + 6.

Finally, simplify the equation to get the vertex form: h(t) = -16(t - 3)² + 150.

From this vertex form, h(t) = -16(t - 3)² + 150, we can directly read the vertex coordinates. The vertex is at (3, 150). This means the maximum height is 150 feet, which occurs at time t = 3 seconds. See, we get the same answer as before! But by completing the square, we see the relationship between the squared term and the vertex. This also shows that the method is correct.

Method 3: Using Calculus

For those of you familiar with calculus, here's another cool way to tackle this problem. We can find the maximum height by taking the derivative of the height function, setting it equal to zero, and solving for 't'. The derivative gives us the rate of change of the height with respect to time, which is the velocity. When the velocity is zero, the ball has reached its maximum height and is momentarily at rest before it starts to fall back down.

So, let’s differentiate h(t) = -16t² + 96t + 6 to find h'(t), the velocity function. h'(t) = -32t + 96.

Set h'(t) = 0 and solve for 't': -32t + 96 = 0.

-32t = -96 t = 3 seconds.

As before, the ball reaches its maximum height at t = 3 seconds. Now, substitute t = 3 back into the original height function: h(3) = -16(3)² + 96(3) + 6 h(3) = -144 + 288 + 6 h(3) = 150 feet.

Once again, we get 150 feet! Calculus provides an elegant way to find the maximum height. It leverages the concept of derivatives to identify the point where the rate of change of height is zero, which is the maximum height. Each method, whether it's using the vertex formula, completing the square, or calculus, provides us with a clear path to the solution, emphasizing the versatility of mathematical tools in solving real-world problems. Let’s remember this well, it is a fun journey!

Conclusion: So, How High Does the Ball Go?

So, guys, no matter which method we used – the vertex formula, completing the square, or calculus – we found that the maximum height the football reaches is 150 feet. This means the ball soars to a height of 150 feet above the ground before gravity pulls it back down. This height includes the initial 6 feet, thanks to our starting point. Isn't it amazing how we can predict the behavior of a ball in motion using some simple math equations? This problem beautifully illustrates the power of mathematics in describing and predicting the physical world.

This understanding is not only useful for predicting the flight of a football, but also for understanding other projectile motions, such as the path of a baseball, a rocket, or even a water fountain. The principles remain the same: an initial velocity, the force of gravity, and an initial position, all working together to determine the trajectory of the object. Keep in mind that air resistance can also affect the results, but the basic model we've used here offers a great starting point for understanding these concepts. So next time you see a football soaring through the air, you’ll have a better appreciation for the science behind it!

I hope you enjoyed this journey into finding the maximum height of a football. Math can be fun and help us understand the world around us. Keep exploring and keep asking questions! If you have any questions, feel free to ask. Cheers!