Stone's Fall Distance In 6th Second: Physics Explained

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Hey everyone! Ever wondered how far a stone falls during just one specific second of its fall? Let's dive into a classic physics problem: calculating the distance a stone travels during the 6th second of its fall, assuming it starts from rest and the acceleration due to gravity (g) is 10 m/s². This might sound tricky, but with a little physics knowledge, we can break it down and understand the solution. So, buckle up, and let’s get started!

Understanding the Problem: Free Fall and the 6th Second

Before we jump into the calculations, let's make sure we understand the core concepts. This problem deals with free fall, which is the motion of an object under the influence of gravity only. We're assuming no air resistance in this scenario, which simplifies the problem. Now, the key here is the phrase "during the 6th second." We're not looking for the total distance the stone falls in 6 seconds, but the distance it covers specifically between the 5th and 6th second. This distinction is crucial for solving the problem correctly. In physics, free fall is a fundamental concept that describes the motion of an object when gravity is the only force acting upon it. To truly grasp the complexities of this concept, it's essential to delve into the underlying principles and equations that govern it. One of the key aspects of free fall is the constant acceleration due to gravity, denoted as 'g,' which is approximately 9.8 m/s² on the Earth's surface. This means that the velocity of an object in free fall increases by 9.8 meters per second every second. To understand the dynamics of free fall, physicists use equations of motion that relate displacement, initial velocity, final velocity, acceleration, and time. These equations, derived from classical mechanics, allow us to predict the position and velocity of an object at any point during its free fall. In this context, the initial conditions, such as the object's starting height and velocity, play a critical role in determining its trajectory. Moreover, the mass of the object does not affect its acceleration in free fall, assuming negligible air resistance. This principle, known as the equivalence principle, is a cornerstone of general relativity, highlighting the intricate relationship between gravity and inertia. Therefore, when analyzing free fall scenarios, we can focus on the gravitational acceleration and time elapsed to calculate the object's motion accurately.

The Physics Equations We'll Use

To solve this, we'll primarily use one of the equations of motion for uniformly accelerated motion. Specifically, we'll use the equation that relates distance (s), initial velocity (u), time (t), and acceleration (a):

s = ut + (1/2)at²

Where:

  • s = distance traveled
  • u = initial velocity
  • t = time
  • a = acceleration (in this case, g = 10 m/s²)

Since the stone starts from rest, our initial velocity (u) is 0. This simplifies the equation quite a bit. Remember, guys, physics isn't just about memorizing equations; it's about understanding when and how to apply them. Choosing the right equation is half the battle! For example, the equation s = ut + (1/2)at² is a cornerstone in physics, enabling us to describe the motion of objects under constant acceleration. Each variable within this equation plays a pivotal role in our understanding. The term 's' represents the displacement or the distance traveled by the object, while 'u' denotes the initial velocity, which is the speed and direction the object has at the start of its motion. The variable 't' stands for the time elapsed during the motion, and 'a' is the constant acceleration experienced by the object. This equation is derived from the fundamental principles of kinematics, which describes the motion of points, objects, and systems of objects without considering the forces that cause the motion. It is based on the assumption that the acceleration is constant and in a straight line. The equation is versatile and can be applied to a wide range of scenarios, from calculating the distance a car travels while accelerating to predicting the trajectory of a projectile. By manipulating the equation and understanding the relationships between its variables, physicists and engineers can solve complex problems and design systems that function according to precise motion requirements. The beauty of s = ut + (1/2)at² lies not only in its simplicity but also in its predictive power, making it an indispensable tool in the field of physics.

Step-by-Step Calculation

Okay, let’s get to the math! Here's how we'll calculate the distance:

  1. Calculate the total distance fallen in 6 seconds: Using the equation s = ut + (1/2)at², with u = 0, a = 10 m/s², and t = 6 s: s₆ = (0)(6) + (1/2)(10)(6²) = 0 + 5 * 36 = 180 meters

  2. Calculate the total distance fallen in 5 seconds: Using the same equation, but with t = 5 s: s₅ = (0)(5) + (1/2)(10)(5²) = 0 + 5 * 25 = 125 meters

  3. Subtract the distance fallen in 5 seconds from the distance fallen in 6 seconds: Distance in the 6th second = s₆ - s₅ = 180 meters - 125 meters = 55 meters

So, the stone falls 55 meters during the 6th second of its fall.

Isn't that neat? We used a simple equation and a bit of logical thinking to solve this problem. The step-by-step calculation not only provides a clear path to the solution but also enhances our understanding of the underlying physics principles. For instance, calculating the total distance fallen in 6 seconds (s₆) and 5 seconds (s₅) separately allows us to appreciate how distance accumulates over time under constant acceleration. The formula s = ut + (1/2)at² is instrumental here, where the initial velocity (u) being zero simplifies the equation, emphasizing the role of acceleration due to gravity (a) and the square of time (t²) in determining the distance. This squared relationship highlights that distance increases quadratically with time, meaning the stone falls much farther in later seconds compared to earlier ones. Subtracting s₅ from s₆ to find the distance traveled specifically during the 6th second provides a practical application of differential thinking, where we isolate a specific time interval to understand the dynamics within it. This method is crucial in many areas of physics and engineering, such as analyzing the motion of projectiles, designing efficient braking systems, or even understanding celestial mechanics. Therefore, this step-by-step approach not only solves the problem but also reinforces crucial problem-solving skills and physical intuitions.

Why This Works: Understanding the Physics Behind It

The reason we subtract the distances is because we're interested in the distance covered only during that 6th second. The stone is constantly accelerating, meaning it falls faster each second. By calculating the total distance at 6 seconds and subtracting the total distance at 5 seconds, we isolate the distance covered specifically during that one-second interval. This method illustrates a fundamental concept in physics: analyzing motion in discrete time intervals to understand instantaneous changes. Consider, for instance, a car accelerating from a standstill. During each second, its speed increases, and the distance it covers in each subsequent second is greater than the previous one. Similarly, the stone in free fall accelerates continuously due to gravity. To determine how far it falls during the 6th second, we need to isolate that specific time frame. The total distance fallen after 6 seconds includes all the previous seconds of falling, while the distance fallen after 5 seconds includes only the first five seconds. By subtracting the latter from the former, we effectively