Determining The Maximum Height Of A Vertically Thrown Body

Introduction: Understanding Projectile Motion and Height Calculation

In the realm of physics, projectile motion stands as a cornerstone concept, describing the trajectory of an object launched into the air under the influence of gravity. Understanding projectile motion is crucial in various fields, from sports science to engineering, as it allows us to predict and analyze the movement of objects in flight. This article delves into a specific scenario: a body thrown vertically upwards. We'll explore how to determine the maximum height reached by this body, given its height equation. The equation provided, h = 20t - (49/10)t^2, describes the height (h) of the body at any given time (t). This equation is a quadratic function, and its graph is a parabola, which is the characteristic path of a projectile under constant gravitational acceleration. Understanding this equation is key to unlocking the secrets of the body's motion, including its maximum height and the time it takes to reach that apex. The coefficients in this equation hold significant physical meanings. The coefficient of the t term (20) represents the initial upward velocity of the body, while the coefficient of the t^2 term (-49/10) is related to the acceleration due to gravity. The negative sign indicates that gravity acts downwards, decelerating the body's upward motion. By carefully analyzing this equation, we can extract valuable information about the body's trajectory and its interaction with gravity. The maximum height represents the peak of the parabola, and our goal is to find the coordinates of this point. This involves using mathematical techniques such as finding the vertex of the parabola, which corresponds to the maximum point on the graph. The vertex not only gives us the maximum height but also the time at which this height is reached. This information is vital in understanding the complete motion of the body, from its initial launch to its eventual return to the ground. In the subsequent sections, we will explore the methods to determine the maximum height, providing a step-by-step guide to solving this type of problem. We will utilize calculus and algebraic approaches to demonstrate the versatility of mathematical tools in physics. By the end of this article, you will have a clear understanding of how to calculate the maximum height of a vertically thrown body and the underlying physics principles that govern its motion. This knowledge will serve as a foundation for further exploration of projectile motion and related concepts.

Mathematical Tools: Calculus and Vertex Form

To determine the maximum height, we can employ two primary mathematical approaches: calculus and the vertex form of a quadratic equation. Calculus offers a powerful method for finding maxima and minima of functions. In this context, we can use differentiation to find the time at which the body reaches its highest point. The first derivative of the height equation with respect to time represents the body's vertical velocity. At the maximum height, the velocity momentarily becomes zero before the body starts descending. Therefore, by setting the first derivative equal to zero and solving for time (t), we can find the time at which the maximum height is reached. Once we have the time, we can substitute it back into the original height equation to calculate the maximum height itself. This method leverages the fundamental principles of calculus, where the derivative provides information about the rate of change of a function. In this case, the rate of change of height with respect to time gives us the velocity, and the point where the velocity is zero corresponds to the maximum height. This approach is particularly useful for understanding the dynamics of motion and how velocity changes over time. Alternatively, we can use the vertex form of a quadratic equation. The height equation h = 20t - (49/10)t^2 is a quadratic equation, and its graph is a parabola. The vertex of this parabola represents the maximum (or minimum) point of the function. The vertex form of a quadratic equation is given by h = a(t - h)^2 + k, where (h, k) are the coordinates of the vertex. The k value represents the maximum height in this case. By converting the given height equation into vertex form, we can directly read off the maximum height. This method involves completing the square, a technique that transforms a quadratic equation into a perfect square trinomial plus a constant term. This algebraic manipulation allows us to rewrite the equation in vertex form, revealing the maximum height and the time at which it occurs. Both calculus and the vertex form provide effective ways to solve this problem. The choice of method often depends on personal preference and the specific context. Calculus offers a more dynamic view of the motion, while the vertex form provides a more direct algebraic approach. In the following sections, we will demonstrate both methods in detail, illustrating their application to the given height equation. By mastering these techniques, you will gain a deeper understanding of how mathematical tools can be used to analyze physical phenomena.

Method 1: Calculus Approach - Finding the Maximum Height

To determine the maximum height using calculus, we start by differentiating the height equation with respect to time. This gives us the velocity equation, which describes how the body's vertical velocity changes over time. The height equation is given by h = 20t - (49/10)t^2. Differentiating this equation with respect to t, we get the velocity equation: v = dh/dt = 20 - (49/5)t. This equation tells us the instantaneous velocity of the body at any time t. The velocity is initially positive, indicating upward motion, but it decreases over time due to the force of gravity. At the maximum height, the body momentarily stops moving upwards before it begins to descend. This means that at the maximum height, the velocity is zero. Therefore, we set the velocity equation equal to zero and solve for t: 0 = 20 - (49/5)t. Solving for t, we get: t = 20 / (49/5) = 100/49 seconds. This is the time at which the body reaches its maximum height. Now that we have the time at which the maximum height is reached, we can substitute this value back into the original height equation to find the maximum height itself. Substituting t = 100/49 into h = 20t - (49/10)t^2, we get: h_max = 20(100/49) - (49/10)(100/49)^2. Simplifying this expression: h_max = 2000/49 - (49/10)(10000/2401) = 2000/49 - 1000/49 = 1000/49 meters. Therefore, the maximum height reached by the body is 1000/49 meters, which is approximately 20.41 meters. This calculus approach provides a clear understanding of the dynamics of the motion. By finding the derivative, we were able to determine the velocity at any given time. Setting the velocity to zero allowed us to pinpoint the exact time at which the body reaches its highest point. This method demonstrates the power of calculus in solving physics problems, especially those involving motion and optimization. The result, 1000/49 meters, represents the peak of the body's trajectory and is a crucial parameter in understanding its overall motion.

Method 2: Vertex Form Approach - Unveiling the Maximum Height Algebraically

An alternative method to determine the maximum height involves using the vertex form of a quadratic equation. This approach is purely algebraic and provides a direct way to find the maximum value of the height function. The given height equation is h = 20t - (49/10)t^2. To rewrite this equation in vertex form, we need to complete the square. First, we rearrange the equation to have the t^2 term first: h = -(49/10)t^2 + 20t. Next, we factor out the coefficient of the t^2 term, which is -49/10: h = -(49/10)(t^2 - (200/49)t). Now, we complete the square inside the parentheses. To do this, we take half of the coefficient of the t term, which is -200/49, and square it. Half of -200/49 is -100/49, and squaring it gives us (100/49)^2 = 10000/2401. We add and subtract this value inside the parentheses: h = -(49/10)(t^2 - (200/49)t + 10000/2401 - 10000/2401). Now, we can rewrite the expression inside the parentheses as a perfect square trinomial: h = -(49/10)((t - 100/49)^2 - 10000/2401). Next, we distribute the -49/10 back into the parentheses: h = -(49/10)(t - 100/49)^2 + (49/10)(10000/2401). Simplifying the constant term, we get: h = -(49/10)(t - 100/49)^2 + 1000/49. This is the vertex form of the quadratic equation, h = a(t - h')^2 + k', where the vertex is (h', k'). In this case, the vertex is (100/49, 1000/49). The k' value, which is 1000/49, represents the maximum height. Therefore, the maximum height reached by the body is 1000/49 meters, which is approximately 20.41 meters. This result matches the one we obtained using the calculus approach. The vertex form method provides a purely algebraic way to find the maximum height, without the need for differentiation. By completing the square, we transformed the equation into a form that directly reveals the vertex of the parabola, which corresponds to the maximum height. This method highlights the power of algebraic manipulation in solving physics problems and provides a valuable alternative to calculus-based approaches.

Comparison and Conclusion: Calculus vs. Vertex Form and the Significance of Maximum Height

Both the calculus and vertex form methods successfully determined the maximum height reached by the body, yielding the same result: 1000/49 meters (approximately 20.41 meters). This consistency underscores the reliability of both approaches in solving this type of problem. However, each method offers a unique perspective and set of advantages. The calculus approach, using differentiation, provides a dynamic view of the motion. It allows us to understand how the velocity changes over time and pinpoint the exact moment when the body reaches its maximum height. This method is particularly useful when analyzing more complex projectile motion scenarios, where the velocity and acceleration are not constant. By finding the derivative, we gain insights into the instantaneous rate of change of the height, which is crucial for understanding the body's trajectory. The vertex form method, on the other hand, offers a more direct algebraic solution. By completing the square, we transformed the height equation into a form that directly reveals the vertex of the parabola, which corresponds to the maximum height. This method is efficient and avoids the need for differentiation, making it a valuable tool for solving quadratic equations and optimization problems. The vertex form also provides the time at which the maximum height is reached, as it is the h' coordinate of the vertex. The choice between these two methods often depends on personal preference and the specific context of the problem. Calculus provides a deeper understanding of the motion's dynamics, while the vertex form offers a quicker algebraic solution. In this specific scenario, both methods are equally effective, but in more complex situations, one method might be more advantageous than the other. The maximum height is a significant parameter in understanding projectile motion. It represents the peak of the body's trajectory and is crucial for predicting its range and time of flight. Knowing the maximum height allows us to analyze the energy transformations involved in the motion, from kinetic energy at launch to potential energy at the highest point. This concept is fundamental in various fields, including sports, engineering, and ballistics. In conclusion, determining the maximum height of a vertically thrown body is a classic physics problem that can be solved using both calculus and algebraic techniques. The calculus approach provides insights into the dynamics of motion, while the vertex form offers a direct algebraic solution. The maximum height itself is a crucial parameter in understanding projectile motion and its applications. By mastering these methods and concepts, you will gain a deeper appreciation for the beauty and power of physics in describing the world around us.