Calculate Velocity: 3kg Object Falling 50m

Hey everyone! Let's break down how to calculate the final velocity of an object falling from a height, and then we'll apply it to a specific problem. This is a classic physics problem that combines concepts of potential energy, kinetic energy, and the acceleration due to gravity. So, grab your thinking caps, and let's dive in!

Understanding the Concepts

Before we jump into calculations, let's make sure we're all on the same page with the underlying principles:

• Potential Energy (PE): This is the energy an object has due to its position relative to a reference point. Near the Earth's surface, we usually calculate it as PE = mgh, where 'm' is mass, 'g' is the acceleration due to gravity, and 'h' is the height.
• Kinetic Energy (KE): This is the energy an object has due to its motion. It's calculated as KE = (1/2)mv², where 'm' is mass, and 'v' is velocity.
• Conservation of Energy: In a closed system (and assuming we're neglecting air resistance here), the total energy remains constant. This means potential energy can be converted into kinetic energy, and vice versa.
• Acceleration due to Gravity (g): Near the Earth's surface, this is approximately 9.8 m/s². It means that for every second an object falls, its velocity increases by 9.8 meters per second.

Understanding these concepts is crucial. Potential energy dictates how much energy the object could have based on its height. Kinetic energy describes how much energy it actually has when moving. The conservation of energy principle links the two, suggesting that as the object falls, potential energy transforms into kinetic energy. And let's not forget gravity, which is the engine driving this transformation, accelerating the object downwards. Without gravity, our object would just hang there, suspended in mid-air, with no urge to plummet!

Problem Setup

Now, let's consider the specific problem: a 3 kg body falls from a 50-meter cliff. We're given that the acceleration due to gravity is 9.8 m/s². Our goal is to find the velocity of the body when it hits the ground. We're neglecting air resistance to keep things simple, which means all the potential energy at the top of the cliff will be converted into kinetic energy just before impact.

Solving the Problem

Here’s how we can solve this:

1. Calculate the Initial Potential Energy (PE):

   PE = mgh = (3 kg) * (9.8 m/s²) * (50 m) = 1470 Joules

2. Apply Conservation of Energy:

   At the top of the cliff, the body has potential energy, and at the bottom, it has kinetic energy. Assuming no energy is lost (no air resistance):

   PE (initial) = KE (final)

   1470 J = (1/2) * m * v²

3. Solve for the Final Velocity (v):

   1470 J = (1/2) * (3 kg) * v²

   1470 = 1.5 * v²

   v² = 1470 / 1.5

   v² = 980

   v = √980 ≈ 31.3 m/s

Therefore, the final velocity of the body when it hits the ground is approximately 31.3 m/s. Looking at the options provided, the closest one is:

B) 30 m/s

So, the answer is B) 30 m/s.

A More Detailed Look at Each Step

Let's ensure we've nailed each step in detail. First off, calculating the potential energy. We're essentially figuring out how much 'stored' energy the object has due to its position. The higher it is, the more potential energy it possesses. Think of it like winding up a toy – the more you wind, the more potential energy is stored, ready to be unleashed.

Next, we invoke the conservation of energy. This is the linchpin of our calculation. It allows us to equate the initial potential energy to the final kinetic energy, assuming no losses. It's like saying all the energy the object had at the top gets completely converted into motion at the bottom. Of course, in the real world, air resistance would play a role, but we're keeping things simple for now.

Finally, solving for the final velocity involves a bit of algebraic manipulation. We're rearranging the equation to isolate 'v' (velocity). We divide, take the square root, and voilà, we have our answer. This step translates the energy value into a speed, telling us how fast the object is moving just before impact.

Why This Matters

Understanding these principles isn't just about solving textbook problems. It's about understanding the world around you. Whether you're analyzing the trajectory of a baseball, designing a roller coaster, or simply wondering why things fall, these concepts are fundamental. Plus, mastering these basics sets you up for more advanced topics in physics and engineering.

Additional Considerations

While we've solved the problem, here are a few additional things to consider:

• Air Resistance: In real-world scenarios, air resistance would significantly affect the final velocity. It would act as a force opposing the motion, reducing the acceleration and thus the final speed.
• More Complex Scenarios: If the object were thrown downwards with an initial velocity, we'd need to account for that initial kinetic energy in our calculations.
• Units: Always pay attention to units! Make sure you're using consistent units (meters for distance, kilograms for mass, seconds for time) to get the correct answer.

Conclusion

So, there you have it! Calculating the final velocity of a falling body is a classic physics problem that illustrates the principles of potential energy, kinetic energy, and conservation of energy. By understanding these concepts, you can tackle a wide range of physics problems and gain a deeper understanding of the world around you. Keep practicing, and you'll become a physics pro in no time!