Free Fall Problem Solving Height Calculation In Physics

In the fascinating world of physics, free fall is a fundamental concept that describes the motion of an object solely under the influence of gravity. Understanding free fall allows us to predict and analyze the movement of objects in various scenarios, from a dropped ball to a skydiver leaping from an airplane. This article delves into a specific free fall problem, exploring how to calculate the height from which an object falls, given the distance it covers in the final second of its descent. We'll break down the problem step by step, applying the principles of kinematics to arrive at the solution. This exploration will not only enhance your understanding of free fall but also highlight the practical applications of physics in everyday situations. Let's embark on this journey of discovery and unlock the secrets of free fall.

Problem Statement: Analyzing the Final Second of Free Fall

The problem we're tackling presents a scenario where a body falls freely from a certain height, denoted as 'h'. The intriguing aspect is that the body covers a significant portion of its total distance, specifically ${ \frac{7h}{16} }$, in just the last second of its fall. Given the acceleration due to gravity (g) as 10 m/s², our objective is to determine the actual height 'h' from which the body was dropped. This problem is a classic example of applying kinematic equations to analyze motion under constant acceleration. To solve this, we need to understand the relationship between distance, time, acceleration, and initial velocity in free fall. The challenge lies in using the information about the distance covered in the last second to deduce the total time of fall and, consequently, the total height. We'll need to carefully dissect the problem, break it down into manageable parts, and apply the appropriate equations to unravel the solution. This problem not only tests our understanding of free fall but also our ability to apply physics principles to real-world scenarios.

Breaking Down the Problem: Applying Kinematic Equations

To effectively solve this free fall problem, we'll employ the fundamental equations of kinematics, which govern motion with constant acceleration. These equations provide a mathematical framework to relate displacement, initial velocity, final velocity, acceleration, and time. In this specific case, the relevant kinematic equation is:

${ s = ut + \frac{1}{2}at^2 }$

where:

• s is the displacement (distance covered)
• u is the initial velocity
• t is the time elapsed
• a is the acceleration

In our problem, the initial velocity (u) is 0 m/s since the body starts from rest. The acceleration (a) is the acceleration due to gravity, which is given as 10 m/s². The key to solving this problem is to consider the motion in two parts: the entire fall from height 'h' and the fall just before the last second. By setting up equations for both scenarios and using the given information about the distance covered in the last second, we can create a system of equations that allows us to solve for the unknowns, including the total height 'h'. This approach requires a careful application of the kinematic equation and a bit of algebraic manipulation to arrive at the solution. Let's dive into the steps involved in setting up and solving these equations.

Step-by-Step Solution: Unraveling the Height

Let's denote the total time of fall as 't' seconds. The distance covered in 't' seconds is the total height 'h'. Using the kinematic equation, we have:

${ h = 0 \cdot t + \frac{1}{2} \cdot 10 \cdot t^2 }$

${ h = 5t^2 }$

Now, let's consider the time (t - 1) seconds, which is the time elapsed before the last second. The distance covered in (t - 1) seconds can be calculated using the same kinematic equation:

${ h' = 5(t - 1)^2 }$

We know that the distance covered in the last second is ${ \frac{7h}{16} }$. This means the distance covered in (t - 1) seconds is the total height 'h' minus the distance covered in the last second:

${ h' = h - \frac{7h}{16} = \frac{9h}{16} }$

Now we have two expressions for h':

${ h' = 5(t - 1)^2 }$

${ h' = \frac{9h}{16} }$

Equating these two expressions, we get:

${ 5(t - 1)^2 = \frac{9h}{16} }$

Substitute ${ h = 5t^2 }$ into the equation:

${ 5(t - 1)^2 = \frac{9 \cdot 5t^2}{16} }$

Simplify the equation:

${ (t - 1)^2 = \frac{9t^2}{16} }$

Take the square root of both sides:

${ t - 1 = \frac{3t}{4} }$

Solve for 't':

${ 4(t - 1) = 3t }$

${ 4t - 4 = 3t }$

${ t = 4 }$ seconds

Now that we have the total time of fall, we can calculate the height 'h':

${ h = 5t^2 = 5 \cdot 4^2 = 5 \cdot 16 = 80 }$ meters

Therefore, the height 'h' from which the body fell is 80 meters. This step-by-step solution demonstrates the power of applying kinematic equations and algebraic manipulation to solve physics problems. By carefully breaking down the problem and using the given information, we were able to determine the unknown height. In the next section, we'll discuss the key concepts and takeaways from this problem.

Key Concepts and Takeaways: Mastering Free Fall Problems

This free fall problem serves as an excellent illustration of several key concepts in physics. Firstly, it reinforces the understanding of free fall as motion under constant acceleration due to gravity. The consistent acceleration allows us to apply the kinematic equations, which are the cornerstone of analyzing such motion. Secondly, the problem highlights the importance of breaking down complex problems into smaller, more manageable parts. By considering the motion in two segments – the entire fall and the fall before the last second – we were able to set up a system of equations that led to the solution. Thirdly, the problem emphasizes the significance of understanding the relationships between physical quantities like distance, time, velocity, and acceleration. The kinematic equations provide a mathematical framework for these relationships, allowing us to predict and calculate the motion of objects.

Moreover, this problem underscores the practical application of physics principles in real-world scenarios. Free fall is not just a theoretical concept; it's a phenomenon that occurs all around us, from a dropped object to the motion of projectiles. By mastering the concepts and techniques involved in solving free fall problems, we gain a deeper understanding of the physical world and develop our problem-solving skills. In conclusion, this exploration of a free fall problem not only provides a solution but also reinforces fundamental physics concepts and their applications, paving the way for tackling more complex challenges in the field of mechanics.

Conclusion: The Elegance of Physics in Motion

In conclusion, the problem of determining the height from which a body falls freely, given the distance covered in the last second, is a testament to the elegance and power of physics. By applying the fundamental principles of kinematics and carefully dissecting the problem, we successfully calculated the height 'h' to be 80 meters. This exercise not only provided a concrete answer but also reinforced our understanding of free fall, kinematic equations, and the importance of analytical problem-solving in physics.

Throughout this exploration, we've seen how the seemingly complex motion of a falling object can be described and predicted using simple yet powerful equations. The ability to break down a problem into manageable steps, identify the relevant physical principles, and apply the appropriate mathematical tools is crucial for success in physics and many other scientific disciplines. This problem serves as a valuable learning experience, highlighting the interconnectedness of concepts and the practical applications of theoretical knowledge. As we continue our journey in physics, the lessons learned from this problem will undoubtedly serve as a foundation for tackling more challenging and intriguing questions about the world around us. The beauty of physics lies in its ability to explain and predict the behavior of the universe, and this free fall problem is just one example of that power and elegance.