Rocket Height: When Does It Reach 196 Feet?

Hey guys! Today, we're diving into a classic physics problem involving projectile motion. We've got a rocket being launched into the air, and its height is described by a quadratic equation. Specifically, the height h(t) of the rocket at time t (in seconds) after launch is given by the equation h(t) = 112t - 16t^2. Our mission, should we choose to accept it, is to figure out when the rocket will reach a height of 196 feet. This is a super common type of problem in math and physics, and understanding how to solve it will be incredibly helpful for you. So, buckle up, and let's get started!

Understanding Projectile Motion

Before we jump into the calculations, let's take a moment to understand what's going on with this equation. The equation h(t) = 112t - 16t^2 represents the path of the rocket under the influence of gravity. The first term, 112t, represents the initial upward velocity of the rocket. The second term, -16t^2, represents the effect of gravity pulling the rocket back down. This is a quadratic equation, and its graph is a parabola, which makes sense because the path of a projectile (like our rocket) is a parabolic arc. The negative coefficient in front of the t^2 term tells us that the parabola opens downwards, meaning the rocket will go up, reach a maximum height, and then come back down.

When you're dealing with projectile motion, understanding the physics behind the equation is just as important as the math. Imagine the rocket launching: it starts with a high upward speed, slows down as it fights against gravity, momentarily stops at its peak height, and then accelerates downwards due to gravity. This is precisely what our equation models. The coefficient '112' represents the initial velocity, dictating how forcefully the rocket is launched upwards. The '-16' term, derived from the acceleration due to gravity (approximately -32 feet per second squared, halved because of the kinematic equation used), pulls the rocket back to Earth. The interplay between these forces dictates the rocket's trajectory and ultimately its height at any given time. This interplay is what creates the parabolic path we discussed earlier, a visual representation of the rocket's journey through the air. By grasping this physical context, you're not just solving an equation; you're unraveling the story of the rocket's flight.

Setting Up the Equation

Okay, so we know the height function h(t) = 112t - 16t^2, and we want to find the time t when the height h(t) is equal to 196 feet. To do this, we simply substitute 196 for h(t) in the equation. This gives us the equation 196 = 112t - 16t^2. Now we have a quadratic equation that we need to solve for t. The next step is to rearrange the equation so that it's in the standard quadratic form, which is at^2 + bt + c = 0. This will make it easier to solve using either factoring or the quadratic formula.

Getting the equation into the standard quadratic form is crucial because it sets us up for using well-established solution methods. Think of it as organizing your tools before starting a project. In our case, we need to move all the terms to one side of the equation, leaving zero on the other side. By doing this, we transform the problem into a recognizable format that we can tackle systematically. This step is not just about mathematical neatness; it's about setting a clear path toward the solution. We're essentially preparing the equation to be 'unlocked' using methods like factoring, completing the square, or the quadratic formula, each designed to efficiently find the roots of a quadratic equation. By rearranging, we're not just manipulating symbols; we're structuring the problem in a way that makes the solution accessible.

Solving the Quadratic Equation

To get our equation into the standard form, let's add 16t^2 to both sides and subtract 112t from both sides. This gives us 16t^2 - 112t + 196 = 0. Now we have a quadratic equation in the form at^2 + bt + c = 0, where a = 16, b = -112, and c = 196. Before we jump into factoring or the quadratic formula, it's always a good idea to see if we can simplify the equation. Notice that all the coefficients (16, -112, and 196) are divisible by 4, and even better, they're all divisible by 16! So, let's divide the entire equation by 16. This gives us t^2 - 7t + 12.25 = 0. The numbers are much smaller now, which will make things easier.

Simplifying the equation by dividing through by a common factor is a smart move in problem-solving. It's like decluttering your workspace before you start a detailed task. By reducing the coefficients, we reduce the chances of making mistakes in subsequent calculations and make the numbers more manageable. This step also prepares the equation for easier factoring or application of the quadratic formula, particularly helpful if you're doing calculations by hand. It's a bit like shifting gears in a car; you're optimizing the equation for smoother, more efficient progress towards the solution. Remember, mathematics isn't just about finding the right answer; it's also about choosing the most elegant and effective path to get there. Simplifying the equation demonstrates a keen awareness of mathematical efficiency.

Now, let's try to solve this quadratic equation. We have a couple of options: factoring or using the quadratic formula. Factoring is often quicker if it's possible, but the quadratic formula always works. Let's see if we can factor our equation: t^2 - 7t + 12.25 = 0. We need to find two numbers that multiply to 12.25 and add up to -7. A little thought (or some trial and error) reveals that -3.5 and -3.5 fit the bill perfectly! So, we can factor the equation as (t - 3.5)(t - 3.5) = 0, which is the same as (t - 3.5)^2 = 0.

Choosing the right method to solve the quadratic equation is a crucial decision point. Factoring, when feasible, is often the quicker route, like taking a shortcut you know is clear. However, the quadratic formula is the reliable workhorse, ensuring a solution even when factoring seems daunting, like having a GPS that works everywhere. Recognizing that our equation, t^2 - 7t + 12.25 = 0, has coefficients that lend themselves nicely to factoring (the product of -3.5 and -3.5 yields 12.25, and their sum is -7) allows us to sidestep the more cumbersome quadratic formula. This choice reflects a deep understanding of equation structure and solution strategies. It's not just about getting to the answer; it's about navigating the mathematical terrain efficiently and elegantly.

Finding the Time

From the factored form (t - 3.5)^2 = 0, we can see that the only solution is t - 3.5 = 0, which means t = 3.5. Since we have a squared term, this means there's only one solution, and it occurs at t = 3.5 seconds. This tells us that the rocket reaches a height of 196 feet at exactly 3.5 seconds. Now, let’s think about what this means in the real world. The rocket is going up, reaches 196 feet at 3.5 seconds, and because this is the only solution, this must be the peak of its trajectory. If we had two solutions, it would mean the rocket reaches 196 feet on the way up and again on the way down. But in this case, there's only one time, which indicates the maximum height.

Interpreting the solution in the context of the problem is what truly brings the math to life. Finding t = 3.5 seconds is more than just a number; it's a pinpoint in the rocket's journey. Because our quadratic equation has a single root, we infer that this is the exact moment the rocket reaches its zenith, the apex of its flight where it momentarily pauses before beginning its descent. This understanding transforms the mathematical result into a vivid picture of the rocket's trajectory. If we had found two solutions, we'd know the rocket passes through 196 feet twice: once on the way up and again on the way down. The uniqueness of our solution enriches our understanding, revealing a critical point in the rocket's flight path. It's a powerful reminder that mathematical answers are not abstract; they are keys to understanding real-world phenomena.

Conclusion

So, we've successfully determined that the rocket reaches a height of 196 feet at t = 3.5 seconds after launch. We did this by setting up the equation, rearranging it into standard quadratic form, and then solving for t using factoring. This problem illustrates a common application of quadratic equations in physics and engineering, especially when dealing with projectile motion. Remember, when you're tackling these kinds of problems, it's crucial to understand the underlying concepts (like how gravity affects the rocket's path) and to carefully set up and solve the equations. And always, always think about what your answer means in the real-world context of the problem! Keep practicing, and you'll become a pro at solving these types of problems in no time!

Remember, guys, math isn't just about formulas and calculations; it's about understanding the world around us. By breaking down complex problems into smaller, manageable steps, we can tackle anything. Keep exploring, keep learning, and most importantly, keep asking questions!