Solving For Time Interval When A Ball's Height Is Less Than 17 Feet

Introduction

In this article, we delve into a classic physics problem involving the motion of an object under the influence of gravity. Specifically, we'll analyze the scenario where Arthur drops a ball from an initial height of 81 feet. The height of the ball, denoted by h, at any given time t (in seconds) is described by the equation h = -16t² + 81. Our primary objective is to determine the time interval during which the ball's height is less than 17 feet. This problem is a practical application of quadratic equations and inequalities, fundamental concepts in mathematics and physics. We'll break down the problem step by step, ensuring a clear and comprehensive understanding of the solution.

Understanding the Equation of Motion

The equation h = -16t² + 81 is a quadratic equation that models the height of the ball as a function of time. The term -16t² represents the effect of gravity, where -16 ft/s² is half the acceleration due to gravity (approximately -32 ft/s²). The positive 81 represents the initial height from which the ball is dropped. This equation assumes that air resistance is negligible and that the ball is dropped with no initial velocity. It's crucial to understand that this equation is a simplification of real-world physics, but it provides a good approximation for short drops. We will use this equation to find the specific time interval when the ball's height is below the 17-foot threshold. Understanding the components of the equation is crucial for accurately interpreting the ball's motion. The quadratic nature of the equation indicates that the ball's height decreases non-linearly with time, accelerating as it falls due to gravity. This concept is fundamental to understanding projectile motion and other physics problems.

Setting Up the Inequality

To find the time interval when the height of the ball is less than 17 feet, we need to set up an inequality. We replace h in the equation with 17, resulting in the inequality -16t² + 81 < 17. This inequality represents the condition we need to solve to find the desired time interval. It's important to understand that inequalities are used to express relationships where one quantity is not equal to another, but rather greater than or less than. In this case, we are interested in the times when the ball's height is strictly less than 17 feet. This inequality is a mathematical representation of the physical condition we are investigating. The next step involves solving this inequality to determine the values of t that satisfy the condition. This will give us the time frame during which the ball's height is below 17 feet. The correct setup of the inequality is crucial for obtaining an accurate solution. A slight error in the inequality setup can lead to a completely different answer, emphasizing the importance of careful attention to detail.

Solving the Quadratic Inequality

Now, let's solve the inequality -16t² + 81 < 17. First, we subtract 81 from both sides to isolate the term with t², giving us -16t² < -64. Next, we divide both sides by -16. Remember, when dividing or multiplying an inequality by a negative number, we must reverse the inequality sign. This gives us t² > 4. To find the values of t that satisfy this inequality, we take the square root of both sides. This results in two possible solutions: t > 2 and t < -2. However, since time cannot be negative in this context, we discard the solution t < -2. Therefore, the relevant solution is t > 2. This means that the height of the ball is less than 17 feet after 2 seconds. This process of solving the inequality involves algebraic manipulation and a careful consideration of the properties of inequalities, particularly the rule about reversing the sign when dividing by a negative number. The final solution, t > 2, represents a time interval, indicating that the ball's height remains below 17 feet for all times greater than 2 seconds. It's important to note that the square root operation yields both positive and negative solutions, but the negative solution is discarded due to the physical constraints of the problem.

Interpreting the Solution

The solution t > 2 indicates that the ball's height is less than 17 feet for all times greater than 2 seconds after it is dropped. This means that after the ball has been falling for 2 seconds, it will be below the 17-foot mark. This result aligns with our intuition: as the ball falls, its height decreases over time due to gravity. The 2-second mark represents the point in time when the ball's height crosses the 17-foot threshold. Before 2 seconds, the ball is above 17 feet, and after 2 seconds, it is below 17 feet. This interpretation highlights the physical meaning of the mathematical solution. It connects the algebraic result to the real-world scenario of a ball falling under gravity. The time interval t > 2 provides a clear and concise answer to the problem, specifying the duration during which the ball is at a height of less than 17 feet. Understanding the physical implications of the mathematical solution is crucial for applying these concepts to other problems and scenarios.

Considering the Physical Context

It's essential to consider the physical context of the problem. In this scenario, time t represents the time elapsed since the ball was dropped, and therefore, it cannot be negative. This is a crucial constraint that we must keep in mind when interpreting the solution. The solution t > 2 makes sense in this context because it indicates that the ball's height is less than 17 feet after a certain amount of time has passed. However, the equation h = -16t² + 81 is only valid until the ball hits the ground. To find the total time the ball is in the air, we need to determine when h = 0. Setting the equation to zero, we get 0 = -16t² + 81. Solving for t, we find t = √(81/16) = 9/4 = 2.25 seconds. This means the ball hits the ground after 2.25 seconds. Therefore, the interval for which the ball's height is less than 17 feet is 2 < t ≤ 2.25 seconds. This demonstrates the importance of considering the physical limitations of the model. While the mathematical solution t > 2 is valid in a purely algebraic sense, the physical reality imposes a limit on the time interval. The ball cannot continue falling indefinitely; it will eventually hit the ground. This additional consideration provides a more complete and accurate understanding of the problem.

Conclusion

In conclusion, by analyzing the equation h = -16t² + 81 and solving the inequality -16t² + 81 < 17, we determined that the height of the ball is less than 17 feet for the time interval t > 2 seconds. However, when considering the physical context, we found that the ball hits the ground after 2.25 seconds, making the precise interval 2 < t ≤ 2.25 seconds. This problem demonstrates the application of quadratic equations and inequalities in physics, specifically in modeling the motion of an object under gravity. It also highlights the importance of interpreting mathematical solutions within their physical context to ensure accuracy and relevance. This type of problem-solving approach is fundamental to both mathematics and physics, providing a framework for analyzing and understanding a wide range of real-world phenomena. The combination of algebraic techniques and physical reasoning allows us to gain a deeper insight into the behavior of objects in motion and the principles that govern their movement. This comprehensive analysis not only provides a solution to the specific problem but also illustrates the broader applicability of these mathematical and physical concepts.