Falling Body Physics: Finding The Drop Height

Hey there, physics enthusiasts! Today, we're diving into a classic problem involving free-falling bodies. This is one of those scenarios that always seems to pop up in physics classes, and for good reason! It's a fantastic way to understand the power of kinematics and how gravity works. The core of this problem revolves around figuring out the total height from which an object has fallen, given some crucial information about its final descent. Specifically, we know that a freely falling body covered the last 30 meters of its fall in a mere 0.5 seconds. Our goal? To pinpoint the total height from which it started its plunge. Let's break down this problem step by step, making sure we cover all the necessary physics principles and calculations to reach the final answer. We'll be using the formulas of uniformly accelerated motion, dealing with the constant acceleration due to gravity, and figuring out the initial conditions. This will involve breaking down the problem into smaller parts, finding intermediate values, and then putting it all together to calculate the final answer! Understanding the concepts of kinematics, and how to apply them, is absolutely key here.

Okay, so the scenario is this: a body is in free fall, meaning gravity is the only force acting upon it (we're ignoring air resistance, of course!). We're told that in the final half-second of its fall, it travels a distance of 30 meters. This is super helpful because it gives us a direct link between time, distance, and the motion of the body during that last little stretch. The challenge is that we need to work backward. We don't know the initial velocity of the object when it started its fall, and we don't know the total time it was in the air. We can start by considering the final segment of the fall, using our known values, to calculate the velocity right before it hit the ground. From there, we'll try to determine the total time of the fall. The value of gravitational acceleration is a constant, which we can consider as 9.8 m/s², the common value we use for these types of calculations. This is a good example of the kind of problem that can show how theoretical physics applies to real-world scenarios. We want to start by thinking about what information we have and what we're trying to figure out. Then, we can look for formulas that relate these things. It's a bit like a detective puzzle, where each piece of information brings you closer to the final solution.

Let’s start to get a plan. We can't immediately use a simple formula like distance = velocity * time because the velocity isn't constant during the entire fall. It's constantly increasing due to gravity. This means we'll need to use some equations that account for constant acceleration. This includes things like the displacement equation, or the final velocity equation. Now, we know that the acceleration is 9.8 m/s², and we know the time (0.5 s) and the distance (30 m) for the last segment of the fall. This allows us to work backwards and calculate the initial velocity of that final segment. Once we have the initial velocity of that segment, we can then determine other information about the motion, which includes, most importantly, the time the body has been falling. With that total time in hand, we can then calculate the total height from which the body started its freefall.

Step-by-Step Solution

Alright, let's break this down into digestible steps, so we don’t get lost along the way.

Step 1: Analyze the Last 0.5 Seconds

Let's focus on those last 0.5 seconds of the fall, where the object covers 30 meters. We'll use the following kinematic equation:

d = v₀t + (1/2)at²

Where:

• d = distance (30 m)
• v₀ = initial velocity at the start of the 0.5-second interval (which we need to find)
• t = time (0.5 s)
• a = acceleration due to gravity (9.8 m/s²)

Now, plug in the values and solve for v₀:

30 = v₀(0.5) + (1/2)(9.8)(0.5)² 30 = 0.5v₀ + 1.225 28.775 = 0.5v₀ v₀ = 57.55 m/s

This v₀ is actually the velocity of the body at the moment, which we can call t₁.

Step 2: Total Time Calculation

Now that we know the initial velocity of the last 0.5 seconds of the fall, we can determine the time of the total fall. Let's name the time it takes to reach the velocity v₀ as t₁, and the total time as t. With these two parameters, we can figure out the time of the total fall. We know that the velocity at the start of the free fall is 0 m/s. We can use the formula for final velocity:

v = u + at

Where:

• v is the final velocity (57.55 m/s)
• u is the initial velocity (0 m/s)
• a is the acceleration due to gravity (9.8 m/s²)
• t is the time

57. 55 = 0 + 9.8t

t = 5.87 s

However, this is only the time from the beginning to the velocity v₀. To calculate the total time, we need to add the final 0.5 seconds. Then, the total time will be 6.37 seconds.

Step 3: Calculate the Total Height

We know that the time it took to fall from the start to the end is 6.37 seconds. Now we can use the formula that links the height, the time, and the gravitational acceleration:

d = (1/2)gt²

Where:

• d is the total distance (height, which we need to find)
• g is the acceleration due to gravity (9.8 m/s²)
• t is the total time (6.37 s)

So, d = (1/2)(9.8)(6.37)² d = 199.04

Therefore, the total height from which the body fell is approximately 199.0 meters.

Conclusion

So, after all that calculation, the final answer is 199.0. We broke down a seemingly complex problem into manageable chunks, using our knowledge of kinematics, and we were able to calculate the height. Remember, the key is always to identify the knowns, the unknowns, and then find the appropriate equations that connect them. It might seem daunting at first, but with practice, you'll get the hang of it, guys! Keep practicing, keep questioning, and keep exploring the amazing world of physics! This problem is a great example of how you can use a few key principles to solve real-world problems. Keep up the good work and keep learning!