A rubber ball dropped from 24 m height loses its kinetic energy by 25%. What is the height to which it rebounds?

Options

A. 6 m
B. 12 m
C. 18 m
D. 24 m

Explanation

To solve the problem of how high a rubber ball rebounds after being dropped from a height of 24 m and losing 25% of its kinetic energy, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Potential Energy**:
The initial potential energy (PE) of the ball when dropped from a height \( h = 24 \, \text{m} \) is given by the formula:
\[ PE = mgh \]
where \( m \) is the mass of the ball, \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)), and \( h \) is the height.

2. **Calculating the Kinetic Energy Just Before Impact**:
When the ball reaches the ground, all the potential energy converts to kinetic energy (KE):
\[ KE = mgh = mg \cdot 24 \]

3. **Determining the Loss of Kinetic Energy**:
The problem states that the ball loses 25% of its kinetic energy upon impact. Therefore, the kinetic energy after the impact is:
\[ KE_{\text{after}} = KE - 0.25 \cdot KE = 0.75 \cdot KE \]
Substituting the expression for KE:
\[ KE_{\text{after}} = 0.75 \cdot (mg \cdot 24) \]

4. **Calculating the Remaining Energy**:
The remaining kinetic energy after the impact can be expressed in terms of the potential energy at the rebound height \( h' \):
\[ KE_{\text{after}} = mgh' \]
Setting the two expressions for kinetic energy equal gives:
\[ mgh' = 0.75 \cdot (mg \cdot 24) \]

5. **Cancelling Out Mass and Gravity**:
Since \( m \) and \( g \) are common factors, we can cancel them out:
\[ h' = 0.75 \cdot 24 \]

6. **Calculating the Rebound Height**:
Now, we calculate \( h' \):
\[ h' = 0.75 \cdot 24 = 18 \, \text{m} \]

Thus, the height to which the rubber ball rebounds is **18 meters**.