An Object With A Mass Of 100 Grams (0.100 Kg) Is Dropped From A Certain Height And Has A Velocity Of 60 Meters/second. If The Potential Energy Of This Object Is 179.99 Joules, What Is The Height From Which The Object Was Dropped?

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In this physics problem, we will delve into the concept of potential energy and its relationship to an object's height above the ground. Potential energy, a form of mechanical energy, is the energy possessed by an object due to its position relative to a reference point. Specifically, gravitational potential energy is the energy an object has because of its height above the Earth's surface. The higher an object is, the greater its potential energy. The formula for gravitational potential energy is given by:

Potential Energy (PE) = m * g * h

where:

  • m is the mass of the object in kilograms (kg)
  • g is the acceleration due to gravity, approximately 9.8 meters per second squared (m/s²)
  • h is the height of the object above the reference point in meters (m)

In this scenario, we have an object with a mass of 100 grams (0.100 kg) falling from a certain height and attaining a velocity of 60 meters per second. The potential energy of the object is given as 179.99 joules. Our objective is to determine the initial height from which the object was dropped. To solve this problem, we will utilize the formula for potential energy and rearrange it to solve for the height (h). We will then substitute the given values into the formula and calculate the height. This exercise will provide a practical application of the potential energy concept and demonstrate how it can be used to determine an object's position in a gravitational field.

Problem Statement

We are given an object with a mass of 100 grams (0.100 kg) that is dropped from a certain height. The object reaches a velocity of 60 meters per second during its descent. The potential energy of the object is measured to be 179.99 joules. The task is to calculate the initial height from which the object was dropped. This problem requires us to apply the principles of potential energy and utilize the given information to determine the unknown height. The solution will involve rearranging the potential energy formula and substituting the known values to arrive at the answer. Understanding the relationship between potential energy, mass, gravity, and height is crucial in solving this problem. By accurately applying the relevant formulas and performing the calculations, we can successfully determine the initial height of the object.

Solution

To determine the height from which the object was dropped, we will utilize the formula for gravitational potential energy:

Potential Energy (PE) = m * g * h

Where:

  • PE = 179.99 joules (given)
  • m = 0.100 kg (given)
  • g = 9.8 m/s² (acceleration due to gravity, a constant)
  • h = height (the unknown we need to find)

To find the height (h), we need to rearrange the formula to isolate h on one side:

h = PE / (m * g)

Now, we can substitute the given values into the formula:

h = 179.99 joules / (0.100 kg * 9.8 m/s²)

Perform the calculation:

h = 179.99 / (0.98)

h ≈ 183.66 meters

Therefore, the height from which the object was dropped is approximately 183.66 meters. This calculation demonstrates the relationship between potential energy, mass, and height. The higher the object is, the greater its potential energy due to its position in the gravitational field. By using the potential energy formula and the given values, we have successfully determined the initial height of the object.

Step-by-step calculation

  1. Identify the known variables: We are given the potential energy (PE = 179.99 joules), the mass of the object (m = 0.100 kg), and the acceleration due to gravity (g = 9.8 m/s²). We need to find the height (h).
  2. Write down the formula for potential energy: The formula for gravitational potential energy is PE = m * g * h.
  3. Rearrange the formula to solve for height (h): To isolate h, we divide both sides of the equation by (m * g), resulting in h = PE / (m * g).
  4. Substitute the known values into the rearranged formula: h = 179.99 joules / (0.100 kg * 9.8 m/s²).
  5. Perform the multiplication in the denominator: 0. 100 kg * 9.8 m/s² = 0.98 kg m/s².
  6. Divide the potential energy by the result from step 5: h = 179.99 joules / 0.98 kg m/s².
  7. Calculate the height: h ≈ 183.66 meters.

Answer

The object was dropped from a height of approximately 183.66 meters.

Key Concepts

Potential Energy

Potential energy is the energy stored in an object due to its position or configuration. In the context of this problem, we are dealing with gravitational potential energy, which is the energy an object possesses due to its height above a reference point, typically the ground. The gravitational potential energy of an object is directly proportional to its mass, the acceleration due to gravity, and its height. The formula for gravitational potential energy is:

PE = m * g * h

where:

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

Understanding potential energy is crucial in physics as it helps explain how energy can be stored and converted into other forms of energy, such as kinetic energy. In this problem, the object initially has potential energy due to its height, and as it falls, this potential energy is converted into kinetic energy, which is the energy of motion. The concept of potential energy is fundamental in various fields, including mechanics, thermodynamics, and electromagnetism. It is also essential in practical applications such as designing energy storage systems and analyzing the motion of objects in gravitational fields.

Gravitational Acceleration

Gravitational acceleration, denoted by the symbol 'g', is the acceleration experienced by an object due to the force of gravity. On the surface of the Earth, the standard value of gravitational acceleration is approximately 9.8 meters per second squared (9.8 m/s²). This means that an object falling freely under the influence of gravity will increase its velocity by 9.8 meters per second every second. Gravitational acceleration is a crucial concept in physics, particularly in the study of motion and forces. It plays a significant role in understanding how objects move under the influence of gravity, such as projectiles, falling bodies, and celestial objects. In this problem, we use the value of gravitational acceleration to calculate the potential energy of the object. The potential energy is directly proportional to the gravitational acceleration, which means that a stronger gravitational field will result in a higher potential energy for the same object at the same height. Understanding gravitational acceleration is essential for solving problems related to motion, energy, and forces in a gravitational field.

Units of Measurement

In physics, it is crucial to use consistent units of measurement to ensure accurate calculations. In this problem, we are dealing with the following units:

  • Mass: Measured in kilograms (kg). The given mass of the object is 100 grams, which we convert to kilograms by dividing by 1000: 100 grams / 1000 = 0.100 kg.
  • Height: Measured in meters (m). This is the quantity we are trying to find.
  • Potential Energy: Measured in joules (J). The given potential energy is 179.99 joules.
  • Gravitational Acceleration: Measured in meters per second squared (m/s²). The standard value on Earth is approximately 9.8 m/s².

Using the correct units is essential for the potential energy formula (PE = m * g * h) to yield accurate results. If the units are mixed up, the calculated height will be incorrect. For example, if we had used grams instead of kilograms for the mass, the resulting height would be significantly different. Therefore, it is always important to check and convert units as necessary before performing calculations in physics problems. Understanding and using the correct units of measurement is a fundamental aspect of physics and is crucial for problem-solving and data analysis.

Conclusion

In conclusion, we successfully calculated the height from which an object with a mass of 100 grams and a potential energy of 179.99 joules was dropped. By utilizing the formula for gravitational potential energy, PE = m * g * h, and rearranging it to solve for height (h), we determined that the object was dropped from approximately 183.66 meters. This problem highlights the relationship between potential energy, mass, gravity, and height. It also emphasizes the importance of using consistent units of measurement in physics calculations. Understanding these concepts is crucial for solving a wide range of physics problems related to energy, motion, and forces. The step-by-step approach outlined in the solution provides a clear and concise method for tackling similar problems involving potential energy and height calculations. By mastering these fundamental principles, students can gain a deeper understanding of the physical world and improve their problem-solving skills in physics.

This problem serves as a practical application of the potential energy concept and demonstrates how it can be used to determine an object's position in a gravitational field. The ability to calculate potential energy and its relationship to height is essential in various fields, including engineering, physics, and other scientific disciplines. By understanding the key concepts and applying the appropriate formulas, we can solve complex problems and gain valuable insights into the behavior of objects in gravitational fields. This exercise reinforces the importance of potential energy as a fundamental concept in physics and its practical applications in real-world scenarios.