A Stone Thrown Downward Initial Speed Calculation

In the realm of physics, the motion of objects under the influence of gravity is a fundamental concept. Understanding the principles of kinematics, particularly uniformly accelerated motion, allows us to predict and analyze the trajectory of objects in free fall. This article delves into a classic physics problem involving a stone thrown vertically downwards from a tower, aiming to determine its initial speed upon release. We will employ the equations of motion to solve this problem, providing a step-by-step explanation for clarity.

Imagine a stone being thrown with an initial speed vertically downwards from the summit of a tower that stands tall at 40 meters. This stone, propelled by both its initial velocity and the relentless pull of gravity, strikes the ground in a mere 2 seconds. Given that the acceleration due to gravity (g) is approximately 10 m/s², our mission is to unravel the initial speed with which the stone was launched.

Before we embark on the mathematical journey, let's take a moment to grasp the underlying concepts that govern this scenario. The stone's motion is a symphony of two key factors: its initial downward velocity and the constant acceleration due to gravity. As the stone plummets towards the earth, gravity acts as a relentless accelerator, causing its velocity to increase uniformly over time. This is the essence of uniformly accelerated motion, a cornerstone of classical mechanics.

To decipher the initial speed, we'll rely on one of the fundamental equations of motion. This equation acts as a bridge, connecting the stone's displacement (the height of the tower), its initial velocity (the unknown we seek), the time it takes to fall, and the ever-present acceleration due to gravity. With this equation in hand, we'll be able to dissect the problem and extract the elusive initial speed.

Let's begin by identifying the key players in this problem and assigning them their rightful symbols. We'll denote the initial speed of the stone as 'u', the time it takes to strike the ground as 't', the acceleration due to gravity as 'g', and the displacement (the height of the tower) as 's'. To maintain consistency and clarity, we'll adopt a sign convention where the downward direction is considered positive. This means that the acceleration due to gravity (g) will be a positive value in our calculations.

With our variables neatly defined and our sign convention established, we're ready to embark on the mathematical exploration. We'll carefully substitute the known values into the equation of motion, unraveling the unknown initial speed with each step. This methodical approach will ensure that we arrive at the correct solution, shedding light on the stone's initial velocity.

The equation of motion that will illuminate our path is the second equation of motion, a cornerstone of kinematics. This equation elegantly relates displacement (s), initial velocity (u), time (t), and acceleration (a): s = ut + (1/2)at². This equation is our compass, guiding us through the intricacies of the stone's motion.

In our specific case, 's' represents the height of the tower (40 meters), 'u' is the initial speed we seek, 't' is the time of flight (2 seconds), and 'a' is the acceleration due to gravity (g = 10 m/s²). With these values in hand, we'll substitute them into the equation, transforming it into an algebraic puzzle waiting to be solved. The unknown initial speed will emerge as we carefully manipulate the equation, revealing the stone's initial velocity.

Now, let's embark on the journey of solving the equation. We'll substitute the known values into the equation of motion: s = ut + (1/2)gt². This transforms into 40 = u(2) + (1/2)(10)(2)². With the values in place, we'll simplify the equation, paving the way for isolating the unknown initial speed.

Simplifying the equation, we get 40 = 2u + 20. This equation is now a straightforward algebraic expression. We'll isolate the term containing 'u' by subtracting 20 from both sides, resulting in 20 = 2u. Finally, we'll divide both sides by 2 to unveil the initial speed: u = 10 m/s. The initial speed of the stone, the answer to our quest, is revealed.

After our meticulous calculations, the initial speed of the stone stands revealed: 10 m/s. This means that the stone was thrown downwards with an initial velocity of 10 meters per second. This initial push, combined with the relentless pull of gravity, propelled the stone towards the ground in just 2 seconds.

Our journey through the problem has not only yielded the answer but also illuminated the power of the equations of motion. These equations, the bedrock of kinematics, allow us to dissect the motion of objects and predict their behavior under the influence of forces. With a firm grasp of these principles, we can unravel the mysteries of motion and gain a deeper understanding of the physical world around us.

In conclusion, by applying the principles of uniformly accelerated motion and the relevant equation of motion, we successfully determined that the initial speed of the stone was 10 m/s. This exercise underscores the importance of understanding fundamental physics concepts in solving real-world problems. The ability to analyze motion, predict trajectories, and unravel the forces at play is a testament to the power of physics in our quest to understand the universe.

The correct answer is (4) 10 m/s.

Physics, kinematics, uniformly accelerated motion, equations of motion, initial speed, gravity, displacement, time, problem-solving, stone, tower, vertical motion.

Q: What is the equation of motion used in this problem? A: The equation of motion used is s = ut + (1/2)at², where s is displacement, u is initial velocity, t is time, and a is acceleration.

Q: How does gravity affect the motion of the stone? A: Gravity acts as a constant acceleration, increasing the stone's velocity as it falls.

Q: Why is it important to define a sign convention? A: Defining a sign convention ensures consistency in calculations, especially when dealing with direction-dependent quantities like velocity and acceleration.

Calculate Initial Speed of a Stone Thrown from a Tower Physics Problem