A body is dropped from height 8m. After striking the surface it rises to 6m, the fractional loss in kinetic energy during impact, is (Assuming the frictional resistance to be negligible)

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(5)`
D. `(1)/(7)`

Answer

To solve the problem of finding the fractional loss in kinetic energy during the impact of a body dropped from a height of 8 meters, we can follow these steps:

### Step 1: Calculate the Initial Potential Energy
The body is dropped from a height of 8 meters. The potential energy (PE) at this height can be calculated using the formula:
\[ PE = m \cdot g \cdot h \]
where:
- \( m \) = mass of the body (we will keep it as \( m \) since it will cancel out later),
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( h = 8 \, \text{m} \).

Thus, the initial potential energy at height 8 m is:
\[ PE_{\text{initial}} = m \cdot g \cdot 8 \]

### Step 2: Calculate the Final Potential Energy
After the body strikes the surface, it rises to a height of 6 meters. The potential energy at this height is:
\[ PE_{\text{final}} = m \cdot g \cdot 6 \]

### Step 3: Determine the Initial and Final Kinetic Energy
Just before the body hits the ground, all the potential energy will have converted into kinetic energy (KE). Therefore:
- Initial kinetic energy (just before impact):
\[ KE_{\text{initial}} = PE_{\text{initial}} = m \cdot g \cdot 8 \]
- Final kinetic energy (just after impact):
\[ KE_{\text{final}} = PE_{\text{final}} = m \cdot g \cdot 6 \]

### Step 4: Calculate the Loss of Kinetic Energy
The loss of kinetic energy during the impact can be calculated as:
\[ \text{Loss of KE} = KE_{\text{initial}} - KE_{\text{final}} \]
Substituting the values:
\[ \text{Loss of KE} = (m \cdot g \cdot 8) - (m \cdot g \cdot 6) \]
\[ \text{Loss of KE} = m \cdot g \cdot (8 - 6) = m \cdot g \cdot 2 \]

### Step 5: Calculate the Fractional Loss in Kinetic Energy
The fractional loss in kinetic energy is given by the formula:
\[ \text{Fractional Loss} = \frac{\text{Loss of KE}}{KE_{\text{initial}}} \]
Substituting the values we have:
\[ \text{Fractional Loss} = \frac{m \cdot g \cdot 2}{m \cdot g \cdot 8} \]
The \( m \) and \( g \) cancel out:
\[ \text{Fractional Loss} = \frac{2}{8} = \frac{1}{4} \]

### Conclusion
The fractional loss in kinetic energy during the impact is:
\[ \frac{1}{4} \]