Maximum Height Of Projectile: Solve The Quadratic Equation

by Alex Johnson 59 views

Have you ever wondered how high a ball goes when you throw it straight up in the air? Or perhaps you're dealing with a physics problem that involves a projectile launched upwards? In either case, understanding how to calculate the maximum height of a projectile is crucial. In this article, we'll break down a common problem: a projectile fired straight upward from the ground with an initial speed of 32 feet per second, where its height h in feet after t seconds is given by the function h(t) = -16t² + 32t. We'll explore a step-by-step approach to find the maximum height of the projectile. Let's dive in!

Understanding the Problem: Projectile Motion and Quadratic Functions

Before we jump into the solution, let's make sure we understand the key concepts. This problem deals with projectile motion, which is the motion of an object thrown or fired into the air, subject to gravity. The path the projectile follows is a curve, and in this simplified scenario (ignoring air resistance), that curve is a parabola. The equation h(t) = -16t² + 32t is a quadratic function, and its graph is indeed a parabola. The coefficient of the t² term (-16) is negative, which means the parabola opens downwards. This is important because the highest point on the parabola, called the vertex, represents the maximum height of the projectile.

To really grasp this, think about what the equation is telling us. The height h(t) changes over time t. Initially, the projectile is launched upwards with a certain speed. As time passes, gravity slows it down until it momentarily stops at its highest point before falling back down. The quadratic equation mathematically describes this motion, capturing the effect of gravity pulling the projectile back to Earth. Therefore, finding the maximum height is equivalent to finding the y-coordinate (height) of the vertex of the parabola. This is a common type of problem in physics and mathematics, demonstrating the practical applications of quadratic functions. The ability to analyze such problems is crucial not just for academic success but also for understanding real-world scenarios involving motion and trajectories.

Method 1: Completing the Square

One way to find the maximum height is by completing the square. This technique transforms the quadratic equation into vertex form, which directly reveals the coordinates of the vertex. Let's walk through the steps:

  1. Factor out the coefficient of the t² term: h(t) = -16(t² - 2t)

    We factor out -16 from the first two terms to simplify the expression inside the parentheses. This step is crucial because it allows us to create a perfect square trinomial.

  2. Complete the square inside the parentheses: To complete the square, we take half of the coefficient of the t term (-2), square it ((-1)² = 1), and add it inside the parentheses. However, since we're adding it inside the parentheses, which is being multiplied by -16, we're actually subtracting 16 from the outside, so we need to add 16 to compensate.

    h(t) = -16(t² - 2t + 1) + 16

    Adding and subtracting the same value ensures that we don't change the overall value of the expression. The expression inside the parentheses is now a perfect square trinomial.

  3. Rewrite the expression as a squared term: h(t) = -16(t - 1)² + 16

    The expression t² - 2t + 1 is equivalent to (t - 1)². This is the key step in completing the square, as we've transformed the quadratic into a form that reveals the vertex.

  4. Identify the vertex: The equation is now in vertex form: h(t) = a(t - h)² + k, where (h, k) is the vertex. In our case, the vertex is (1, 16).

The vertex (1, 16) tells us that the maximum height occurs at t = 1 second, and the maximum height is 16 feet. Completing the square is a powerful technique for rewriting quadratic equations and revealing their key properties. It's a fundamental skill in algebra and is widely applicable in various mathematical and scientific contexts. This method provides a clear and direct way to identify the maximum or minimum value of a quadratic function, making it a valuable tool for problem-solving.

Method 2: Using the Vertex Formula

Another method to find the maximum height involves using the vertex formula. For a quadratic function in the form f(x) = ax² + bx + c, the x-coordinate of the vertex is given by x = -b / 2a. In our case, h(t) = -16t² + 32t, so a = -16 and b = 32. Let's apply the formula:

  1. Find the t-coordinate of the vertex: t = -b / 2a = -32 / (2 * -16) = -32 / -32 = 1

    This tells us that the maximum height occurs at t = 1 second, the same result we obtained using completing the square.

  2. Substitute the t-value into the equation to find the maximum height: h(1) = -16(1)² + 32(1) = -16 + 32 = 16

The vertex formula provides a concise and efficient way to find the x-coordinate of the vertex, which in this case represents the time at which the maximum height is reached. By substituting this value back into the original equation, we can directly calculate the maximum height. This method is particularly useful when you only need to find the vertex and don't require the equation to be in vertex form. Understanding and applying the vertex formula is a key skill in analyzing quadratic functions and solving related problems. It complements the completing the square method, providing an alternative approach that can be more convenient in certain situations.

The maximum height of the projectile is 16 feet. This confirms our result from the completing the square method. The vertex formula is a valuable shortcut for finding the maximum or minimum value of a quadratic function without having to go through the steps of completing the square. Both methods provide valuable insights into the behavior of quadratic functions and their applications in real-world problems.

Conclusion: Maximum Height Achieved

Both methods, completing the square and using the vertex formula, lead us to the same conclusion: the maximum height of the projectile is 16 feet. This maximum height is reached at t = 1 second after the projectile is fired. Understanding how to solve these types of problems is fundamental in physics and mathematics, providing a strong foundation for more complex concepts.

The ability to analyze projectile motion and quadratic functions is not just an academic exercise. It has practical applications in various fields, such as sports, engineering, and even video game design. Whether you're calculating the trajectory of a ball, designing a bridge, or creating realistic physics simulations, the principles we've discussed here are essential. By mastering these concepts, you'll not only excel in your studies but also gain a deeper appreciation for the world around you.

For further exploration of projectile motion and quadratic functions, you might find valuable resources on websites like Khan Academy's Algebra I course.