Question

A ball dropped from a height of \(4 \mathrm{~m}\) rebounds to a height of \(2.4 \mathrm{~m}\) after hitting the ground. Then the percentage of energy lost is (A) 40 (B) 50 (C) 30 (D) 600

Short answer

The percentage of energy lost for the ball is 40% when dropped from a height of 4m and rebounds to a height of 2.4m.

Step-by-step solution

Calculate Initial and Final Potential Energies

To find the initial potential energy (PE_initial) and final potential energy (PE_final), we will use the formula for potential energy: \[PE = mgh\] where \(m\) = mass of the ball (unknown in this case, but it will cancel out in the end), \(g\) = acceleration due to gravity (\(9.81 m/s^2\)), \(h\) = height of the ball. PE_initial will be when the ball is dropped from a height of 4m while PE_final will be when the ball rebounds to a height of 2.4m.

Calculate the Energy Loss

To calculate the energy loss, we will subtract the final potential energy (PE_final) from the initial potential energy (PE_initial): \[Energy\,Loss = PE_{initial} - PE_{final}\]

Calculate the Percentage of Energy Lost

To find the percentage of energy lost, we will divide the energy loss by the initial potential energy and then multiply it by 100: \[%Energy\,Lost = \frac{Energy\,Loss}{PE_{initial}} \times 100\] Now let's plug in the values and calculate the percentage of energy lost:

Substitute the values and calculate

Using the potential energy formula, we get: \( \begin{cases} PE_{initial} = m \times 9.81 \times 4 \\ PE_{final} = m \times 9.81 \times 2.4 \end{cases} \) Therefore, the energy loss is: \[Energy\,Loss = (m \times 9.81 \times 4 ) - (m \times 9.81 \times 2.4)\] Now we will calculate the percentage of energy lost: \[%Energy\,Lost = \frac{(m \times 9.81 \times 4 ) - (m \times 9.81 \times 2.4)}{m \times 9.81 \times 4} \times 100\] Simplifying the equation, we get: \[%Energy\,Lost = \frac{9.81m (4 - 2.4)}{9.81m \times 4} \times 100\] Notice that the mass "m" and the acceleration due to gravity "9.81" cancel out: \[%Energy\,Lost = \frac{(4 - 2.4)}{4} \times 100\] Thus, \[%Energy\,Lost = (1 - 0.6) \times 100 = 0.4 \times 100\] \[%Energy\,Lost = 40\%\] So, the percentage of energy lost for the ball is 40% which corresponds to option (A).

Key concepts

Understanding Gravitational Potential Energy

The concept of gravitational potential energy (GPE) is fundamental when considering objects at a height above the ground. It represents the energy that an object possesses because of its position in a gravitational field. The formula for calculating gravitational potential energy is \( PE = mgh \), where \( m \) is the mass of the object, \( g \) is the acceleration due to gravity (approximately \( 9.81 \frac{m}{s^2} \) on Earth), and \( h \) is the height from which the object is elevated.

When a ball is dropped from a certain height, it possesses a certain amount of gravitational potential energy. This energy is fully based on the height from which it is dropped. For our ball dropped from a height of 4 meters, it initially has potential energy proportional to 4 meters. When it rebounds to 2.4 meters, the potential energy is less, proportional to this lower height.

In these problems, it is common to not focus on the actual numerical value of \( m \), because it often cancels out during calculations. This simplification makes it easier to focus on changes in energy rather than absolute values.

Energy Loss in Physical Processes

Whenever an object like a ball bounces, some of its mechanical energy is lost. In a perfect system with no external forces, energy levels would remain constant, but in reality, factors like air resistance and internal friction cause energy dissipation.

When the ball hits the ground, some of its gravitational potential energy doesn't convert into kinetic energy but is lost as sound, heat, or deformation energy when it impacts the surface. This is why the ball rebounds to a lower height compared to its initial drop height. The energy loss can be calculated by the difference between the initial and final gravitational potential energies. For our specific problem:

• Initial energy is calculated at 4 meters height.
• Final energy is calculated at 2.4 meters height.
• The difference is the energy lost in the process.

Understanding this concept is crucial for diving deeper into energy conservation and loss in real-world situations.

Calculating the Percentage of Energy Lost

To quantify how much energy a system loses, we often use a percentage calculation. This involves comparing the energy lost to the initial energy of the system. These calculations help in determining the efficiency of energy transfer processes.

The percentage of energy lost is calculated using the formula:
\[\% \text{Energy Lost} = \frac{\text{Energy Loss}}{\text{Initial Potential Energy}} \times 100\]
The calculation is straightforward if you follow these steps:

• Determine the energy lost by subtracting the final potential energy from the initial potential energy.
• Divide the energy loss by the initial energy to standardize the value.
• Multiply by 100 to convert the fraction into a percentage.

In our ball example, this results in the formula \( \% \text{Energy Lost} = \frac{(4 - 2.4)}{4} \times 100 \), which simplifies to a 40% energy loss. This means 40% of the ball's initial potential energy wasn’t recovered when it bounced.