Hammer Impact: Height After Energy Loss

Let's dive into a physics problem involving a hammer, energy, and a bit of calculation! We've got a 10 kg hammer falling from a height of 2 meters. Upon impact, it absorbs 150 J of energy. The question is: how high could the hammer rise on the opposite side after the impact, considering the energy loss? Sounds interesting, right? Let's break it down and figure out the correct answer together.

Understanding the Problem

When we talk about the hammer falling and rising, we're really dealing with the concepts of potential energy and energy conservation. Initially, the hammer has potential energy due to its height. As it falls, this potential energy converts into kinetic energy. Upon impact, some of that energy is lost (in this case, 150 J is absorbed). The remaining energy will determine how high the hammer can bounce back up. Understanding these principles is key to solving the problem accurately. So, before jumping into calculations, make sure you're comfortable with the basics of energy conservation and potential energy. It will make the whole process much smoother and easier to grasp. Trust me, a solid understanding of the underlying concepts makes physics problems way less intimidating!

Initial Potential Energy

Okay, so let's figure out the initial potential energy (PE) of our hammer before it even starts falling. The formula for potential energy is PE = mgh, where:

• m = mass (in kg)
• g = acceleration due to gravity (approximately 9.8 m/s²)
• h = height (in meters)

In our case:

• m = 10 kg
• g = 9.8 m/s²
• h = 2 m

Plugging these values into the formula, we get:

PE = 10 kg * 9.8 m/s² * 2 m = 196 J

So, the hammer starts with 196 Joules of potential energy. This is the total energy the hammer has before anything happens. Remember this number, because it's going to be important when we figure out how high the hammer can bounce back up after it loses some energy on impact.

Energy After Impact

Now, here comes the part where we account for the energy lost during the impact. The problem tells us that the hammer absorbs 150 J of energy when it hits the ground. This means that out of the initial 196 J of potential energy, only a portion remains to propel the hammer back up. To find out how much energy is left, we simply subtract the absorbed energy from the initial potential energy:

Remaining Energy = Initial Potential Energy - Absorbed Energy Remaining Energy = 196 J - 150 J = 46 J

So, after the impact, the hammer has only 46 Joules of energy left. This remaining energy will be converted back into potential energy as the hammer rises on the opposite side. Keep this value in mind, as it will help us determine the maximum height the hammer can reach after the impact.

Calculating the Height

Alright, now for the final step: calculating the height the hammer can reach after the impact. We know the remaining energy (46 J) and we want to find the new height (h). We'll use the same potential energy formula, but this time we'll solve for h:

PE = mgh 46 J = 10 kg * 9.8 m/s² * h

To isolate h, we divide both sides of the equation by (10 kg * 9.8 m/s²):

h = 46 J / (10 kg * 9.8 m/s²) h = 46 / 98 h ≈ 0.469 meters

Therefore, the hammer could rise to approximately 0.469 meters on the opposite side after the impact. This is the height the hammer can reach with the remaining energy after losing 150 J upon impact. So, based on our calculations, the closest answer from the options provided would be 0.5 meters.

Analyzing the Options

Now that we've done the math, let's take a look at the answer choices and see which one aligns with our calculation. We found that the hammer could rise to approximately 0.469 meters, which is very close to 0.5 meters. So, let's examine each option:

• a) 0.5 meters: This is very close to our calculated value of 0.469 meters, making it the most likely answer.
• b) 1 meter: This is significantly higher than our calculated value, suggesting that it's not the correct answer.
• c) 1.5 meters: This is even higher than 1 meter, further deviating from our calculated value. It's unlikely to be the correct answer.
• d) 2 meters: This is the initial height, but we know that the hammer loses energy upon impact, so it cannot reach the same height again. This option is incorrect.

Based on our analysis, the closest and most plausible answer is a) 0.5 meters. This makes sense considering the energy loss upon impact, which reduces the height the hammer can reach on the opposite side.

Final Answer

So, putting it all together, the final answer is:

a) 0.5 meters

Conclusion

In summary, we determined that a 10 kg hammer dropped from 2 meters, absorbing 150 J of energy upon impact, could rise to approximately 0.5 meters on the opposite side. We arrived at this conclusion by first calculating the initial potential energy of the hammer, then subtracting the energy lost during impact, and finally using the remaining energy to calculate the new height. Remember, the key to solving such problems lies in understanding the principles of energy conservation and potential energy. Make sure to practice more problems to solidify your understanding. Keep up the great work!