Determining Time Interval For Projectile Height Exceedance A Detailed Solution

In the realm of physics and mathematics, projectile motion stands as a cornerstone concept, particularly when dissecting the trajectory of objects launched into the air. One classic problem involves a projectile fired vertically upwards, its height varying with time due to gravity's influence. This article delves into such a problem, providing a step-by-step solution and offering insights into the underlying principles.

Problem Statement Decoding the Scenario

At the heart of our exploration lies the following scenario: A projectile is launched directly upwards from the ground with an initial velocity of 112 feet per second (ft/s). The height, denoted as h, of the projectile above the ground at time t seconds is described by the equation h = -16t² + 112t. Our mission is to determine the time interval during which the projectile's height surpasses 160 feet. This involves understanding the interplay between initial velocity, gravity, and the resulting parabolic trajectory.

Understanding Projectile Motion The Physics Behind the Problem

To effectively tackle this problem, it's crucial to grasp the fundamentals of projectile motion. When an object is projected upwards, it experiences a constant downward acceleration due to gravity, approximately 32 ft/s². This acceleration causes the object's upward velocity to decrease until it momentarily reaches zero at its highest point, after which it begins to descend. The equation h = -16t² + 112t mathematically models this motion, where the -16t² term represents the effect of gravity and the 112t term accounts for the initial upward velocity. The height of the projectile at any given time is a crucial aspect of understanding its trajectory, and this equation allows us to calculate that height. We need to understand how to solve this quadratic equation to find the specific times when the projectile reaches a certain height.

Setting Up the Inequality Defining the Target Height

The core of the problem lies in identifying the time interval during which the projectile's height, h, exceeds 160 feet. Mathematically, we express this condition as an inequality: -16t² + 112t > 160. This inequality forms the foundation for our solution, guiding us to find the values of t that satisfy this condition. Understanding how to manipulate and solve this inequality is paramount to determining the time frame in which the projectile is above the specified height.

Solving the Inequality A Step-by-Step Approach

To solve the inequality -16t² + 112t > 160, we embark on a series of algebraic manipulations aimed at isolating the variable t. This process involves transforming the inequality into a more manageable form, allowing us to identify the range of time values that meet the specified height condition. Each step is crucial in accurately determining the solution set.

Transforming the Inequality Moving Terms to One Side

The initial step in solving the inequality involves moving all terms to one side, setting the stage for factoring or applying the quadratic formula. Subtracting 160 from both sides of the inequality, we obtain: -16t² + 112t - 160 > 0. This transformation consolidates the terms, making it easier to identify the coefficients and constants necessary for further steps in the solution process. This rearrangement is a common technique in solving inequalities and equations, streamlining the subsequent algebraic manipulations.

Simplifying the Inequality Dividing by a Common Factor

To simplify the inequality, we identify and divide by a common factor, making the coefficients smaller and the inequality easier to work with. In this case, we observe that all terms are divisible by -16. Dividing both sides by -16 (and remembering to reverse the inequality sign because we're dividing by a negative number), we get: t² - 7t + 10 < 0. This simplification not only makes the numbers more manageable but also reveals the underlying quadratic structure of the inequality, setting the stage for factoring or using the quadratic formula to find the solution.

Factoring the Quadratic Expression Unveiling the Roots

The simplified quadratic expression, t² - 7t + 10, can be factored to reveal its roots, which are critical points in determining the solution to the inequality. Factoring involves expressing the quadratic as a product of two binomials. In this case, we find that t² - 7t + 10 can be factored into (t - 2)(t - 5). This factorization is a key step in solving quadratic equations and inequalities, as it transforms the problem into a simpler form where we can easily identify the values of t that make the expression equal to zero.

Identifying the Critical Points Defining the Intervals

The factored form, (t - 2)(t - 5) < 0, allows us to identify the critical points where the expression equals zero. These critical points are t = 2 and t = 5. These values divide the time axis into three intervals: t < 2, 2 < t < 5, and t > 5. Understanding these intervals is crucial because the sign of the quadratic expression will remain constant within each interval. By testing a value within each interval, we can determine whether the expression is positive or negative in that interval, allowing us to identify the solution set for the inequality.

Testing Intervals Determining the Solution Set

To determine the solution set, we test a value within each interval to see if it satisfies the inequality (t - 2)(t - 5) < 0. For t < 2, let's test t = 0. The expression becomes (0 - 2)(0 - 5) = 10, which is not less than 0. For 2 < t < 5, let's test t = 3. The expression becomes (3 - 2)(3 - 5) = -2, which is less than 0. For t > 5, let's test t = 6. The expression becomes (6 - 2)(6 - 5) = 4, which is not less than 0. Therefore, the inequality is satisfied only when 2 < t < 5. This interval represents the times when the projectile's height exceeds 160 feet.

Solution and Interpretation Decoding the Time Interval

Based on our analysis, the solution to the inequality -16t² + 112t > 160 is 2 < t < 5. This means that the projectile's height exceeds 160 feet during the time interval between 2 seconds and 5 seconds after it is fired. This interval provides a clear understanding of when the projectile reaches and surpasses the specified height threshold. Understanding the meaning of this interval in the context of the problem is crucial for a complete solution.

Interpreting the Solution Contextualizing the Result

The time interval 2 < t < 5 provides valuable information about the projectile's trajectory. It tells us that the projectile reaches a height of 160 feet at t = 2 seconds on its way up and then descends back to 160 feet at t = 5 seconds. Therefore, the projectile spends 3 seconds (from t = 2 to t = 5) above the height of 160 feet. This interpretation provides a practical understanding of the projectile's motion, highlighting the relationship between time, height, and the influence of gravity. It's a crucial step in translating a mathematical solution into a real-world context.

Verification of Solution Checking the Answer

To ensure the accuracy of our solution, it's essential to verify the result. We can do this by plugging in values within and outside the interval 2 < t < 5 into the original inequality. For example, let's test t = 3 (within the interval): -16(3)² + 112(3) = -144 + 336 = 192, which is greater than 160. Now let's test t = 1 (outside the interval): -16(1)² + 112(1) = -16 + 112 = 96, which is not greater than 160. This verification process confirms that our solution is correct, providing confidence in the accuracy of our analysis.

Conclusion Mastering Projectile Motion Problems

In conclusion, we have successfully determined the time interval during which the projectile's height exceeds 160 feet. By setting up and solving the inequality -16t² + 112t > 160, we found that the projectile is above this height between 2 seconds and 5 seconds after being fired. This problem serves as a valuable example of how mathematical principles can be applied to understand and analyze real-world phenomena like projectile motion. Understanding the relationship between initial conditions, gravity, and time allows us to predict and interpret the behavior of objects in motion.

Key Takeaways Practical Insights from the Solution

This problem highlights several key takeaways for solving projectile motion problems. First, setting up the correct equation or inequality based on the problem statement is crucial. Second, algebraic manipulation, such as simplifying and factoring, is essential for solving the equation or inequality. Third, understanding the physical interpretation of the solution is vital for connecting the mathematical result to the real-world scenario. Finally, verifying the solution ensures accuracy and builds confidence in the result. These key takeaways provide a framework for tackling similar problems in physics and mathematics.

Further Exploration Expanding Knowledge and Skills

To further enhance your understanding of projectile motion, consider exploring related concepts such as the maximum height reached by the projectile, the time it takes to reach the ground, and the effect of air resistance on the trajectory. Additionally, practice solving similar problems with varying initial velocities and height thresholds. This continuous learning and practice will solidify your understanding of projectile motion and improve your problem-solving skills. Engaging with different scenarios and challenges will deepen your grasp of the concepts and enhance your ability to apply them in various contexts.