Free Fall Problem: Find The Initial Height Of A Stone

Hey guys! Ever wondered how to calculate the initial height of an object in free fall? This is a classic physics problem that combines the principles of kinematics and gravity. Today, we're going to break down a specific scenario step-by-step. We will find out how to determine the initial height from which a stone was dropped, given that it travels 100 meters in the last 2 seconds of its free fall, with the acceleration due to gravity (g) being 10 m/s². Understanding free fall is crucial in physics. It's a concept that explains the motion of objects under the influence of gravity alone, neglecting air resistance. This involves understanding concepts such as initial velocity, acceleration due to gravity, time, and displacement. When an object is in free fall, it experiences constant acceleration downwards, which on Earth is approximately 9.8 m/s², often rounded to 10 m/s² for simpler calculations. This constant acceleration means the object's velocity increases uniformly over time. To solve problems related to free fall, we often use kinematic equations, which are mathematical expressions that describe the motion of objects with constant acceleration. These equations relate displacement, initial velocity, final velocity, acceleration, and time. By applying these equations, we can determine various aspects of an object's motion, such as the distance it falls, its velocity at a particular time, or the time it takes to fall a certain distance. In real-world scenarios, air resistance can significantly affect the motion of falling objects, but in many introductory physics problems, we ignore air resistance to simplify the calculations and focus on the fundamental principles of free fall. So, let's dive into the problem and see how we can apply these concepts to find the solution!

Breaking Down the Problem

Let's dissect the question. We know the stone travels 100 meters in the final 2 seconds of its fall, and the acceleration due to gravity (g) is 10 m/s². Our mission is to find the initial height from which the stone was dropped. To solve this, we'll use the equations of motion, specifically those related to free fall. These equations help us relate displacement, initial velocity, final velocity, acceleration, and time. Understanding these relationships is key to cracking this problem. One of the first things to consider in this type of problem is the reference frame. We need to define a starting point and direction for our measurements. In this case, it's convenient to consider the initial position of the stone (where it was dropped) as the reference point (0 meters) and the downward direction as positive. This simplifies the calculations and helps avoid confusion with signs. Next, we need to identify the knowns and unknowns. We know the distance traveled in the last 2 seconds (100 meters), the time interval (2 seconds), and the acceleration due to gravity (10 m/s²). The unknowns include the initial height, the total time of fall, and the velocity of the stone just before it entered the final 2 seconds of its fall. By carefully listing these knowns and unknowns, we can choose the appropriate kinematic equations to solve for the unknowns. It’s also helpful to visualize the problem. Imagine the stone falling from a height, accelerating constantly due to gravity. The final 2 seconds are just a part of the entire journey, but they provide crucial information for us to work backward and find the initial height. So, let's get started with the calculations and find the answer together!

Solving for the Unknowns

First, let's consider the last 2 seconds of the fall. We can use the equation of motion: d = v₀t + (1/2)gt², where d is the distance (100 m), v₀ is the initial velocity at the beginning of those 2 seconds, t is the time (2 s), and g is the acceleration due to gravity (10 m/s²). Plugging in the values, we get: 100 = 2v₀ + (1/2)10(2²). Solving for v₀, we find the velocity of the stone at the start of the last 2 seconds. This velocity becomes our key to unlocking the earlier stages of the fall. Now, let's calculate v₀: 100 = 2v₀ + 20. This simplifies to 80 = 2v₀, so v₀ = 40 m/s. This means that at the beginning of the last 2 seconds, the stone was already moving at a speed of 40 m/s. This is an important piece of information that will help us determine the total time of fall and the initial height. Next, we need to figure out how long it took the stone to reach this velocity of 40 m/s from the moment it was dropped. We can use another equation of motion: v = v₀ + gt, where v is the final velocity (40 m/s), v₀ is the initial velocity (0 m/s, since the stone was dropped), g is the acceleration due to gravity (10 m/s²), and t is the time. Plugging in the values, we can solve for t, which will give us the time it took to reach 40 m/s. This time, combined with the last 2 seconds, will give us the total time of fall. So, let's calculate the time it took to reach 40 m/s.

Calculating Time and Total Height

Using the equation v = v₀ + gt, we have 40 = 0 + 10t. Solving for t, we get t = 4 seconds. This means it took 4 seconds for the stone to reach a velocity of 40 m/s. Now, we know the stone fell for 4 seconds before entering the final 2 seconds, making the total time of fall 6 seconds (4 seconds + 2 seconds). With the total time of fall, we can now calculate the total distance the stone fell, which will give us the initial height. To find the total height, we'll use the equation d = v₀t + (1/2)gt² again, but this time, we'll use the total time of fall (6 seconds) and the initial velocity at the very beginning (0 m/s, since the stone was dropped). Plugging in these values, we'll find the total distance the stone fell, which is the initial height we're looking for. So, let's do the calculation! Using d = v₀t + (1/2)gt², we have d = 0*(6) + (1/2)10(6²). This simplifies to d = (1/2)1036, which equals 180 meters. Therefore, the initial height from which the stone was dropped is 180 meters. This completes our calculation, and we have successfully found the answer to the problem! It's amazing how we can use the laws of physics to understand and predict the motion of objects around us. Understanding the concepts and applying the correct equations are key to solving these types of problems. So, let's move on to the final answer and wrap things up.

Final Answer

So, the stone was dropped from a height of 180 meters. That's option C) in the original question. We successfully used the equations of motion and a bit of logical thinking to solve this problem. Remember, the key to these physics problems is to break them down into smaller, manageable steps. Identify what you know, what you need to find, and then choose the right equations to connect the dots. Guys, I hope this explanation helped you understand how to tackle free fall problems. Physics can be super interesting once you grasp the fundamentals! Understanding the principles of free fall and how to apply kinematic equations is a valuable skill, not just for physics students, but for anyone interested in the world around them. These concepts are used in various fields, from engineering to sports, and even in everyday situations where we observe the motion of objects. By mastering these fundamentals, you'll be able to analyze and predict the motion of objects in free fall with confidence. Remember, practice makes perfect! The more you work through problems like this, the more comfortable you'll become with the equations and the concepts. Don't be afraid to break down complex problems into smaller steps, identify the knowns and unknowns, and choose the appropriate equations to solve for the unknowns. And most importantly, have fun exploring the world of physics!

Answer: C) 180 m