Calculate Building Height And Stone Impact Velocity After 5 Second Drop

Hey guys! Ever wondered how we can figure out the height of a tall building just by dropping a stone from its top? Or how fast that stone is traveling when it hits the ground? It's all thanks to the magic of physics, and we're going to break it down in a way that's super easy to understand. Let's dive into this fascinating problem together!

Understanding the Problem: Stone Dropped from a Building

So, we've got this classic physics scenario: a stone is dropped from the top of a building, and it takes 5 seconds to hit the ground. Our mission is twofold. First, we need to calculate the height of the building. Second, we want to figure out the magnitude of the velocity – basically, how fast the stone is going – when it finally crashes into the ground. This involves understanding the principles of free fall and using some basic kinematic equations.

Let's get this straight, guys. When we talk about an object falling freely, we're diving into the world of constant acceleration. Now, the magic ingredient here is gravity! Near the Earth's surface, gravity is this constant force that pulls everything downwards, making things accelerate at roughly 9.8 meters per second squared (9.8 m/s²). Think of it like this: every second an object falls, it gets faster by 9.8 meters per second. Cool, right? Now, here's the kicker: we're going to ignore air resistance for simplicity's sake. Yeah, in the real world, air pushes back on falling stuff, but for this problem, we're imagining a nice, clean, vacuum-like scenario. This lets us focus purely on the effect of gravity, making our calculations a whole lot easier. So, with gravity as our constant companion, we can use some snazzy equations to figure out the height and impact velocity of our falling stone. Ready to roll?

Determining the Height of the Building

Now, let's figure out how tall this building actually is! We're going to use a super handy kinematic equation that relates distance, initial velocity, time, and acceleration. This equation is like our secret weapon for solving free fall problems. Remember, we're trying to find the distance (which is the height of the building in this case), and we know the time (5 seconds) and the acceleration due to gravity (9.8 m/s²). The equation we're going to use is: d = v₀t + (1/2)at²

Where:

• d is the distance (the height of the building we want to find).
• v₀ is the initial velocity (how fast the stone was moving when it was first dropped).
• t is the time the stone takes to fall (5 seconds).
• a is the acceleration due to gravity (9.8 m/s²).

Okay, so before the stone took its nosedive, it was just chilling at the top of the building, right? That means its initial velocity (v₀) was zero. Zip. Nada. Zilch. This is a super important detail because it simplifies our equation big time! So now our equation looks like this:

d = (0)t + (1/2)at²

See how that v₀t term just vanishes? Awesome! Now we're cooking with gas. Let's plug in the values we know:

d = (1/2) * 9.8 m/s² * (5 s)²

First, let's square that 5 seconds: 5 squared is 25. So now we have:

d = (1/2) * 9.8 m/s² * 25 s²

Next, let's multiply 9.8 by 25, which gives us 245:

d = (1/2) * 245 m

Finally, let's take half of 245. We end up with 122.5 meters:

d = 122.5 m

Boom! There you have it, folks. The height of the building is a whopping 122.5 meters. That's pretty tall! We used the power of physics and a bit of algebra to figure that out. Not too shabby, eh?

Calculating the Impact Velocity

Alright, now that we know how tall the building is, let's tackle the next part of our problem: figuring out how fast the stone was zooming when it hit the ground. This is what we call the impact velocity, and we've got another nifty kinematic equation to help us out. Get ready to use this equation:

v = v₀ + at

Where:

• v is the final velocity (the impact velocity we're trying to find).
• v₀ is the initial velocity (remember, the stone started from rest, so it's 0 m/s).
• a is the acceleration due to gravity (still 9.8 m/s²).
• t is the time the stone takes to fall (our trusty 5 seconds).

Just like before, the initial velocity v₀ is zero because the stone started from rest. This makes our equation even simpler:

v = 0 + at

Which is the same as:

v = at

Awesome! Now, let's plug in those values:

v = 9.8 m/s² * 5 s

Time for some multiplication! 9. 8 multiplied by 5 gives us 49:

v = 49 m/s

There we have it! The stone hits the ground with a velocity of 49 meters per second. That's seriously fast! To put that into perspective, 49 meters per second is roughly 176 kilometers per hour (or about 109 miles per hour). Imagine the stone is like a tiny speeding bullet when it makes impact. It's all thanks to gravity constantly accelerating it downwards for those 5 seconds. This calculation shows how powerful seemingly simple physics principles can be in describing the world around us.

Putting It All Together

So, guys, we've successfully tackled a classic physics problem! We figured out two important things about a stone dropped from a building: how tall the building is and how fast the stone is traveling when it hits the ground. Here's a quick recap:

• The height of the building: Using the equation d = v₀t + (1/2)at², we plugged in our values (initial velocity = 0, time = 5 seconds, acceleration = 9.8 m/s²) and calculated the height to be 122.5 meters. That's a pretty tall structure!
• The impact velocity: We used the equation v = v₀ + at (which simplified to v = at because the initial velocity was zero) and found that the stone hits the ground with a velocity of 49 meters per second. That's seriously speedy!

These calculations demonstrate the power of kinematic equations in understanding motion, especially in situations involving constant acceleration like free fall. We've seen how gravity plays a crucial role in making things fall faster and faster. By understanding these basic principles, we can predict and explain the motion of objects in our world. It's pretty amazing stuff when you think about it.

Real-World Applications and Further Exploration

Okay, guys, so we've solved this cool problem about a stone falling from a building, but you might be thinking,