Calculating Potential Energy Of A Sled A Physics Problem

When dealing with physics problems, understanding the concepts and formulas is crucial. This article focuses on potential energy, a fundamental concept in mechanics. We'll explore how to calculate the potential energy of a sled at the top of a slope, using the formula $PE = mgh$. This formula is a cornerstone in understanding how energy is stored due to an object's position within a gravitational field. Let's delve deeper into potential energy and its calculation with a practical example.

Defining Potential Energy

Potential energy, often abbreviated as PE, is the energy an object possesses due to its position relative to a force field. In simpler terms, it’s the energy stored in an object because of its height or position. For instance, a book held above the ground has potential energy due to its height. When released, this potential energy converts into kinetic energy as the book falls. The greater the height, the more potential energy the object has, and consequently, the more kinetic energy it can gain. Understanding this relationship between potential and kinetic energy is vital in physics.

The Formula: PE = mgh

The formula to calculate gravitational potential energy is $PE = mgh$, where:

• PE represents potential energy, measured in joules (J).
• m denotes the mass of the object, measured in kilograms (kg).
• g stands for the acceleration due to gravity, approximately 9.8 m/s² on Earth.
• h represents the height of the object above a reference point, measured in meters (m).

This formula highlights that potential energy is directly proportional to the mass of the object, the acceleration due to gravity, and the height. A heavier object at the same height will have more potential energy than a lighter one. Similarly, an object at a greater height will possess more potential energy than the same object at a lower height. The gravitational constant, g, ensures that the units align correctly, resulting in energy measured in joules.

Problem Statement: Sled at the Top of a Slope

Let's consider the problem at hand: A sled with a mass of 45 kg is at rest at the top of a slope that is 2 m high. The question is: What is the sled's potential energy? To solve this, we will use the potential energy formula and plug in the given values. This step-by-step calculation will illustrate how to apply the formula effectively and arrive at the correct answer. Let’s break down the known values and then perform the calculation.

Identifying the Given Values

Before we start calculating, let's identify the values provided in the problem:

• Mass of the sled (m): 45 kg
• Height of the slope (h): 2 m
• Acceleration due to gravity (g): Approximately 9.8 m/s²

Now that we have all the necessary values, we can proceed to calculate the potential energy. This structured approach ensures that we don't miss any critical information and can apply the formula accurately. The next step is to substitute these values into the $PE = mgh$ equation.

Calculation: Applying the Formula

Now, let's plug the values into the potential energy formula:

$PE = mgh$ $PE = (45 \text{ kg}) \times (9.8 \text{ m/s}^2) \times (2 \text{ m})$

By performing this calculation, we will determine the potential energy of the sled at the top of the slope. This is a straightforward application of the formula, emphasizing the importance of accurate substitution and multiplication. Let’s go through the arithmetic step by step to ensure clarity and precision.

Step-by-Step Calculation

First, multiply the mass (45 kg) by the acceleration due to gravity (9.8 m/s²):

$45 \text{ kg} \times 9.8 \text{ m/s}^2 = 441 \text{ kg m/s}^2$

Next, multiply the result by the height (2 m):

$441 \text{ kg m/s}^2 \times 2 \text{ m} = 882 \text{ kg m}^2\text{/s}^2$

Since 1 joule (J) is equal to 1 kg m²/s², the potential energy is:

$PE = 882 \text{ J}$

Thus, the potential energy of the sled at the top of the slope is 882 joules. This result highlights how the potential energy is derived from the sled’s mass and height, influenced by Earth's gravitational pull.

Answer: Potential Energy of the Sled

The calculated potential energy of the sled is 882 J. Therefore, the correct answer from the given options is:

D. 882 J

This answer confirms that the sled has a potential energy of 882 joules due to its position at the top of the 2-meter slope. This energy is stored and ready to be converted into kinetic energy once the sled starts moving down the slope. Understanding this conversion is a key aspect of studying energy in physics.

Analyzing the Result

The potential energy of 882 J represents the amount of energy the sled possesses due to its position in the gravitational field. This energy is waiting to be transformed into kinetic energy as the sled moves down the slope. The magnitude of the potential energy is influenced by the sled’s mass and its height above the reference point. A higher slope or a heavier sled would result in even greater potential energy. This understanding is fundamental in various physics applications, from simple mechanics to more complex systems.

Conclusion: Key Takeaways on Potential Energy

In conclusion, we've explored the concept of potential energy and how to calculate it using the formula $PE = mgh$. We've applied this formula to a practical example of a sled at the top of a slope, demonstrating the step-by-step process of identifying given values, substituting them into the formula, and calculating the potential energy. The key takeaways from this discussion are:

• Potential energy is the energy an object has due to its position.
• The formula $PE = mgh$ is used to calculate gravitational potential energy.
• Potential energy depends on the object's mass, height, and the acceleration due to gravity.
• Understanding potential energy is crucial in various physics applications and real-world scenarios.

By mastering this concept, you'll be well-equipped to tackle more complex problems involving energy and motion. Remember to always identify the given values, apply the formula correctly, and analyze the results to gain a deeper understanding of the underlying principles.