Calculating Maximum Height In Vertical Projectile Motion A Physics Guide

In the fascinating realm of physics, understanding projectile motion is crucial, especially when analyzing objects moving under the influence of gravity. Let's delve into the specifics of a body thrown vertically upwards, exploring the equation that governs its motion and how to determine the maximum height it reaches. This article aims to provide a comprehensive explanation of the concepts involved, making it accessible to both students and enthusiasts alike. We will dissect the given equation, apply relevant physics principles, and employ mathematical techniques to arrive at the solution. By the end of this exploration, you will have a solid understanding of how to calculate the maximum height of a projectile launched vertically upwards, along with the underlying physics principles that govern its trajectory.

The Equation of Motion: Deciphering the Trajectory

The motion of the body is described by the equation:

$$h = 20t - \frac{49}{10}t^2$$

Where:

• h represents the height attained by the body at time t.
• t represents the time elapsed since the body was thrown upwards.

The coefficients in this equation hold significant physical meanings. The term 20t indicates the initial upward velocity of the body. If there were no gravitational acceleration, the body would continue to rise indefinitely at a constant rate of 20 units of height per unit of time. However, the term -\frac{49}{10}t^2 accounts for the effect of gravity, which acts to decelerate the body as it ascends. The negative sign indicates that gravity opposes the upward motion, and the coefficient \frac{49}{10} (which is 4.9) is related to the acceleration due to gravity, approximately 9.8 m/s², halved because this equation describes displacement, not velocity. This quadratic relationship signifies that the body's upward velocity decreases linearly with time due to gravity, eventually reaching zero at the maximum height.

The equation is a quadratic equation in the form of h = at^2 + bt + c, where a = -\frac{49}{10}, b = 20, and c = 0. The parabolic nature of this equation indicates that the body's trajectory is a parabola, symmetrical about the vertical line passing through the vertex. The vertex of this parabola represents the maximum height attained by the body, and the corresponding time value represents the time at which this maximum height is reached. Understanding the components of this equation is the first step in determining the maximum height, and it highlights the interplay between initial velocity and gravitational acceleration in determining the projectile's path.

Determining the Maximum Height: Methods and Approaches

To find the maximum height reached by the body, we can employ two primary methods: completing the square and using calculus. Both approaches leverage the properties of quadratic equations and their graphical representation as parabolas. The maximum height corresponds to the vertex of the parabola, which represents the highest point on the trajectory. By finding the coordinates of this vertex, we can determine both the time at which the maximum height is reached and the maximum height itself. Understanding both methods provides a robust toolkit for solving similar problems and deepens the understanding of the underlying mathematical principles.

Method 1: Completing the Square

Completing the square is an algebraic technique used to rewrite a quadratic equation in a form that reveals the vertex of the parabola. This method involves manipulating the equation to create a perfect square trinomial, which can then be easily factored. By isolating the squared term, we can identify the vertex coordinates directly. This method is particularly useful for visualizing the transformations of the parabola and understanding how the coefficients of the quadratic equation affect its shape and position. The steps involved in completing the square are as follows:

1. Rewrite the equation: Start with the equation h = 20t - \frac{49}{10}t^2 and rearrange it as h = -\frac{49}{10}t^2 + 20t. This puts the equation in the standard quadratic form, making it easier to manipulate.
2. Factor out the coefficient of t^2: Factor out -\frac{49}{10} from the terms involving t, resulting in h = -\frac{49}{10}(t^2 - \frac{200}{49}t). This step isolates the quadratic and linear terms within the parentheses, preparing them for the completion of the square.
3. Complete the square: To complete the square, take half of the coefficient of the t term inside the parentheses (which is -\frac{100}{49}), square it (resulting in (\frac{100}{49})^2), and add and subtract it inside the parentheses. This gives h = -\frac{49}{10}(t^2 - \frac{200}{49}t + (\frac{100}{49})^2 - (\frac{100}{49})^2). The addition and subtraction of the same term do not change the value of the expression but allow us to create a perfect square trinomial.
4. Rewrite as a squared term: Rewrite the perfect square trinomial as a squared term: h = -\frac{49}{10}((t - \frac{100}{49})^2 - (\frac{100}{49})^2). This step consolidates the quadratic and linear terms into a single squared term, revealing the vertex form of the equation.
5. Distribute and simplify: Distribute the -\frac{49}{10} and simplify the equation: h = -\frac{49}{10}(t - \frac{100}{49})^2 + \frac{1000}{49}. This final form of the equation is the vertex form, which directly reveals the coordinates of the vertex.

From this form, we can see that the vertex of the parabola is at the point (\frac{100}{49}, \frac{1000}{49}). Therefore, the maximum height reached by the body is \frac{1000}{49} units.

Method 2: Using Calculus (Finding the Vertex)

Calculus provides a powerful tool for finding the maximum or minimum of a function. In this case, we want to find the maximum height, which corresponds to the maximum value of the function h(t) = 20t - \frac{49}{10}t^2. The key concept here is that at the maximum point, the derivative of the function (which represents the slope of the tangent line) is zero. This is because the function changes from increasing to decreasing at a maximum point, and the tangent line becomes horizontal.

1. Find the derivative: The first step is to find the derivative of h(t) with respect to t. This gives us the rate of change of height with respect to time, which is the vertical velocity of the body. Using the power rule of differentiation, we get h'(t) = 20 - \frac{49}{5}t. The derivative represents the instantaneous velocity of the projectile at any time t.
2. Set the derivative to zero: To find the time at which the height is maximum, we set the derivative equal to zero and solve for t: 20 - \frac{49}{5}t = 0. This equation represents the condition where the vertical velocity is zero, which occurs at the maximum height. Solving for t gives t = \frac{100}{49}.
3. Find the second derivative (optional): To confirm that this value of t corresponds to a maximum (rather than a minimum or an inflection point), we can find the second derivative of h(t). The second derivative is h''(t) = -\frac{49}{5}, which is negative. A negative second derivative indicates that the function is concave down, meaning that the critical point is indeed a maximum. While not strictly necessary for finding the maximum height, the second derivative test provides a rigorous confirmation of the nature of the critical point.
4. Substitute t back into the original equation: Substitute the value of t we found (t = \frac{100}{49}) back into the original equation for h(t) to find the maximum height: h(\frac{100}{49}) = 20(\frac{100}{49}) - \frac{49}{10}(\frac{100}{49})^2. This step calculates the height of the projectile at the time when its vertical velocity is zero, which is the maximum height reached.
5. Simplify: Simplify the expression to find the maximum height: h(\frac{100}{49}) = \frac{2000}{49} - \frac{1000}{49} = \frac{1000}{49}. This final calculation gives the maximum height reached by the projectile, which is the same result obtained using the method of completing the square.

Both methods lead to the same result, reinforcing the understanding of the physics and mathematics involved. The choice of method often depends on personal preference and the specific context of the problem. Completing the square provides a visual and algebraic approach, while calculus offers a more general method applicable to a wider range of functions.

The Maximum Height: Interpreting the Result

Both methods, completing the square and calculus, yield the same result: the maximum height reached by the body is \frac{1000}{49} units. This value is approximately 20.41 units. This result has significant physical meaning. It represents the highest point the body reaches in its trajectory before gravity begins to pull it back down. The maximum height is a crucial parameter in projectile motion, as it helps to characterize the overall trajectory and the influence of initial velocity and gravitational acceleration.

Furthermore, the time at which the maximum height is reached, t = \frac{100}{49}, is approximately 2.04 seconds. This value represents the time it takes for the body to reach its highest point. The total time of flight (the time it takes for the body to return to its initial position) would be twice this value, assuming negligible air resistance and a flat trajectory. This symmetry is a characteristic feature of projectile motion under constant gravitational acceleration.

The maximum height is directly influenced by the initial upward velocity and the acceleration due to gravity. A higher initial velocity will result in a greater maximum height, while a stronger gravitational field will reduce the maximum height. This relationship is evident in the equation of motion, where the initial velocity term contributes positively to the height, while the gravitational term contributes negatively. Understanding these relationships allows us to predict and control the trajectory of projectiles in various scenarios.

Real-World Applications and Implications

The principles of projectile motion and maximum height calculation have numerous real-world applications. From sports like basketball and baseball, where the trajectory of the ball is crucial for success, to engineering applications such as designing artillery projectiles and launching satellites, understanding these concepts is essential. In sports, athletes intuitively account for projectile motion when throwing or hitting a ball, adjusting the angle and velocity to achieve the desired range and height. Engineers use these principles to design systems that launch objects into specific trajectories, optimizing for factors such as distance, height, and accuracy.

In addition, the concepts of projectile motion are fundamental in fields such as ballistics and aerospace engineering. Ballistics experts use these principles to analyze the trajectories of bullets and other projectiles, while aerospace engineers use them to design spacecraft trajectories and calculate orbital parameters. The ability to accurately predict and control the motion of projectiles is critical in these fields, and a deep understanding of the underlying physics is essential for success.

The study of projectile motion also has implications for understanding natural phenomena, such as the movement of volcanic ejecta or the trajectories of meteoroids entering the Earth's atmosphere. By applying the principles of physics and mathematics, we can gain insights into these complex processes and make predictions about their behavior. This knowledge is valuable for mitigating potential hazards and for expanding our understanding of the natural world.

Conclusion: Mastering Projectile Motion

In conclusion, determining the maximum height of a body thrown vertically upwards involves understanding the interplay between initial velocity, gravitational acceleration, and the equation of motion. By employing methods such as completing the square and calculus, we can accurately calculate the maximum height and gain a deeper understanding of the projectile's trajectory. This knowledge has wide-ranging applications in sports, engineering, and the study of natural phenomena.

The equation h = 20t - \frac{49}{10}t^2 provides a mathematical framework for analyzing the motion of the body, and the maximum height represents a crucial parameter that characterizes the trajectory. By mastering the concepts and techniques presented in this article, you can confidently solve similar problems and apply your knowledge to real-world scenarios. The study of projectile motion is a fascinating and rewarding endeavor, offering insights into the fundamental principles that govern the motion of objects in our universe. The ability to accurately predict and control the motion of projectiles is a valuable skill, and a deep understanding of the underlying physics is essential for success in a variety of fields.

Vertical Projectile Motion, Maximum Height, Equation of Motion, Completing the Square, Calculus, Quadratic Equation, Gravity, Initial Velocity, Trajectory, Vertex