Problem 57 - College Physics

Question

\(\bullet\) A 12.0 g plastic ball is dropped from a height of 2.50 \(\mathrm{m}\) and is moving at 3.20 \(\mathrm{m} / \mathrm{s}\) just before it hits the floor. How much mechanical energy was lost during the ball's fall?

Step-by-Step Solution

Answer

The mechanical energy lost is 0.23256 J.

Step 1: Calculate Initial Potential Energy

The initial potential energy, when the ball is at the height of 2.50 m, can be calculated using the formula \(PE_{initial} = m \cdot g \cdot h\), where \(m = 0.012\,\mathrm{kg}\) is the mass, \(g = 9.8\,\mathrm{m/s^2}\) is the acceleration due to gravity, and \(h = 2.5\,\mathrm{m}\) is the height. Compute \(PE_{initial} = 0.012 \cdot 9.8 \cdot 2.5\).

Step 2: Calculate Initial Potential Energy Value

Substituting the values into the equation, we get \(PE_{initial} = 0.294\, \mathrm{J}\). This represents the potential energy the ball has at the start.

Step 3: Calculate Final Kinetic Energy

The kinetic energy of the ball just before it hits the ground can be calculated using the formula \(KE_{final} = \frac{1}{2} m v^2\), where \(m = 0.012\,\mathrm{kg}\) and \(v = 3.20\,\mathrm{m/s}\). Compute \(KE_{final} = 0.5 \cdot 0.012 \cdot (3.20)^2\).

Step 4: Calculate Final Kinetic Energy Value

Substitute the values into the kinetic energy formula to get \(KE_{final} = 0.06144\, \mathrm{J}\). This is the energy just before the ball hits the floor.

Step 5: Determine the Energy Loss

The mechanical energy lost during the fall can be determined by finding the difference between the initial potential energy and the final kinetic energy. Compute \(\text{Energy Lost} = PE_{initial} - KE_{final}\), which is \(0.294\, \mathrm{J} - 0.06144\, \mathrm{J}\).

Step 6: Calculate the Amount of Energy Lost

After performing the subtraction, the energy lost is \(0.23256\, \mathrm{J}\). This represents the mechanical energy converted into other forms, such as sound or heat, during the fall.

Key Concepts

Potential Energy

Potential energy is the energy stored in an object due to its position or height. When we talk about potential energy in physics, we're often referring to gravitational potential energy, which depends on the height of an object above a reference point, like the ground. To calculate gravitational potential energy, we use the formula:\[PE = m \cdot g \cdot h\]Where:

• \(m\) is the mass of the object (in kilograms)
• \(g\) is the acceleration due to gravity (approximately \(9.8 \, \text{m/s}^2\) on Earth)
• \(h\) is the height above the reference point (in meters)

In the original problem, by substituting the values \(m = 0.012 \, \text{kg}\), \(g = 9.8 \, \text{m/s}^2\), and \(h = 2.50 \, \text{m}\) into the formula, we found that the initial potential energy was \(0.294 \, ext{J}\). This energy is an indication of what could be converted into other forms as the ball begins its descent.

Kinetic Energy

Kinetic energy is the energy of motion. When an object is moving, it possesses kinetic energy, which we can calculate using the formula:\[KE = \frac{1}{2} m v^2\]Where:

• \(m\) is the mass of the object (in kilograms)
• \(v\) is the velocity of the object (in meters per second)

In the given problem, the plastic ball acquires kinetic energy as it falls towards the ground. By substituting the values \(m = 0.012 \, \text{kg}\) and \(v = 3.20 \, \text{m/s}\) into the kinetic energy formula, we calculated the final kinetic energy as \(0.06144 \, ext{J}\). This energy increases as the ball speeds up during its fall, marking the conversion from potential to kinetic energy as height decreases.

Energy Conversion

Energy conversion involves changing energy from one form to another. In physics, specifically in this exercise, we observe how potential energy is transformed into kinetic energy as the object falls. During this transformation, not all the potential energy becomes kinetic energy. In real-world scenarios, not all energy is perfectly converted into another form due to energy losses, often resulting in sound, heat, or deformation energy.
In this problem, the difference between the initial potential energy (\(0.294 \, ext{J}\)) and the final kinetic energy (\(0.06144 \, ext{J}\)) indicates that some energy was lost. The mechanical energy lost, which sums up to \(0.23256 \, ext{J}\), is what likely dissipated into heat through air resistance or sound when the ball hit the ground.
Understanding energy conversion mechanisms helps explain the different outcomes of energy processes and allows us to calculate how efficiently energy is used or transformed in physical systems.

Physics Problem Solving

Physics problem solving involves breaking down a problem into smaller, more manageable parts to find a solution. It typically involves:

• Identifying the known and unknown variables
• Using relevant physics formulas
• Substituting given data into equations
• Carrying out calculations carefully to find results
• Interpreting the results to understand the physical situation

In the exercise shown, our approach started by calculating potential and kinetic energies separately. Then, finding the difference gave us insight into energy lost as non-mechanical energy, such as sound or heat.
By systematically working through problems step-by-step and applying the basic principles of physics, students can better understand how different forms of energy interact. This enables them to solve more complex scenarios beyond textbook exercises.