Hillside Height Calculation: Potential Energy & Dog (40kg)

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Hillside Height Calculation: Potential Energy & Dog (40kg)

Hey guys! Let's break down this physics problem together. We've got a scenario where a 40 kg dog is chilling at the top of a hillside, possessing a potential energy of 1,568 J. Our mission, should we choose to accept it (and we do!), is to figure out the height of that hillside. Sounds like fun, right? The formula we'll be using is PE = mgh, where PE stands for potential energy, m is mass, g is the acceleration due to gravity, and h is the height we're trying to find. So, let's dive in and see how we can solve this.

Understanding Potential Energy

First off, let's make sure we're all on the same page about potential energy. In simple terms, potential energy is the energy an object has because of its position relative to a reference point. Think of it like this: the higher up something is, the more potential energy it has because gravity has a longer distance to pull it down. In our case, the dog at the top of the hill has potential energy because of its height above the ground. This energy is just waiting to be converted into kinetic energy (the energy of motion) if the dog decides to take a tumble down the hill! When dealing with potential energy near the Earth's surface, we're usually talking about gravitational potential energy, which is what our formula, PE = mgh, calculates. The ‘m’ represents the mass of the object (in kilograms), ‘g’ is the acceleration due to gravity (approximately 9.8 m/s² on Earth), and ‘h’ is the height (in meters), which is what we're after. So, now that we've refreshed our understanding of potential energy, let's see how we can use this knowledge to solve our problem and find the height of the hillside.

Applying the Formula: PE = mgh

Okay, let's get down to brass tacks and apply the formula PE = mgh to our problem. We know the dog's mass (m) is 40 kg, the potential energy (PE) is 1,568 J, and the acceleration due to gravity (g) is approximately 9.8 m/s². What we don't know, and what we're trying to find, is the height (h). So, we need to rearrange the formula to solve for h. A little bit of algebra magic gives us: h = PE / (mg). Now, it's just a matter of plugging in the values we know. We have h = 1568 J / (40 kg * 9.8 m/s²). Let's break that down. First, we multiply the mass (40 kg) by the acceleration due to gravity (9.8 m/s²), which gives us 392 kgm/s². Then, we divide the potential energy (1568 J) by this result (392 kgm/s²). Remember that a Joule (J) is equivalent to a kg*m²/s², so the units will work out nicely, leaving us with meters (m), which is exactly what we want for height. Now, let's do the division: 1568 / 392 = 4. So, we've found that h = 4 meters. This means the height of the hillside is 4 meters. Pretty cool, huh? We've used a simple formula and some basic physics principles to solve a real-world problem. Now, let's consider the answer choices and pick the correct one.

Solving for Height: Step-by-Step

Let's walk through the process of solving for the height (h) step-by-step, just to make sure everything is crystal clear. Remember, our formula is PE = mgh, and we want to isolate h.

  1. Write down the formula: Start by writing down the formula we're using: PE = mgh.
  2. Rearrange the formula: To solve for h, we need to get it by itself on one side of the equation. To do this, we divide both sides of the equation by mg: h = PE / (mg)
  3. Plug in the values: Now, we substitute the values we know into the equation. We have PE = 1,568 J, m = 40 kg, and g = 9.8 m/s². So, our equation becomes: h = 1568 J / (40 kg * 9.8 m/s²)
  4. Calculate the denominator: First, we multiply the mass (40 kg) by the acceleration due to gravity (9.8 m/s²): 40 kg * 9.8 m/s² = 392 kg*m/s²
  5. Divide: Now, we divide the potential energy (1568 J) by the result from the previous step (392 kgm/s²): h = 1568 J / 392 kgm/s² = 4 m
  6. State the answer: We've found that the height h is 4 meters. So, the height of the hillside is 4 meters.

By breaking it down into these steps, you can see exactly how we arrive at the answer. It's all about understanding the formula, rearranging it correctly, plugging in the known values, and then doing the math. With a little practice, you'll be solving these types of problems like a pro! Next up, we'll take a look at the answer choices provided and see which one matches our calculated height.

Identifying the Correct Answer

Alright, we've crunched the numbers and found that the height of the hillside is 4 meters. Now, let's take a look at the answer choices provided in the question and see which one matches our result. The options are:

A. 3.9 m B. 4.0 m C. 39.2 m D. 40.0 m

Comparing our calculated height of 4 meters with the answer choices, we can see that option B, 4.0 m, is the correct answer. It matches our calculation perfectly! Options A, C, and D are incorrect. Option A is close, but not quite the right answer, while options C and D are significantly off. So, the correct answer is B. 4.0 m. This means the dog is sitting atop a hillside that is 4 meters high. We successfully used the potential energy formula to calculate the height, and then we identified the correct answer choice. Great job, guys! We've tackled this physics problem head-on and come out victorious. Remember, understanding the concepts and breaking down the problem into manageable steps is key to solving these types of questions. Now, let's wrap things up with a final recap and some key takeaways.

Final Answer and Key Takeaways

So, to recap, we were given a scenario where a 40 kg dog had 1,568 J of potential energy while sitting on a hillside, and we needed to find the height of that hillside. We used the formula PE = mgh, rearranged it to solve for height (h), plugged in the given values, and calculated the height to be 4 meters. Therefore, the correct answer is B. 4.0 m.

Here are some key takeaways from this problem:

  • Understanding Potential Energy: Potential energy is the energy an object has due to its position relative to a reference point. In this case, it's the energy the dog has because of its height above the ground.
  • The Formula PE = mgh: This formula is crucial for calculating gravitational potential energy. Make sure you understand what each variable represents (PE for potential energy, m for mass, g for acceleration due to gravity, and h for height).
  • Rearranging Formulas: Being able to rearrange formulas is a fundamental skill in physics and math. Practice isolating the variable you're trying to find.
  • Step-by-Step Approach: Breaking down a problem into smaller, manageable steps can make it much easier to solve. Write down the formula, rearrange it, plug in the values, do the calculations, and state the answer.
  • Units: Always pay attention to the units! In this problem, we were working with Joules (J) for energy, kilograms (kg) for mass, meters per second squared (m/s²) for acceleration due to gravity, and meters (m) for height. Making sure your units are consistent is essential for getting the correct answer.

Great job, everyone! We successfully solved this problem and learned some valuable physics concepts along the way. Keep practicing, and you'll become a physics whiz in no time!