A ball bounce of `80%` of its original height . What fraction of its mechanical energy is lost in each bounce ?

Options

A. `0.20`
B. `0.60`
C. `0.40`
D. `1`

Explanation

To solve the problem of determining the fraction of mechanical energy lost in each bounce of a ball that bounces to 80% of its original height, we can follow these steps:

### Step 1: Understand the Initial and Rebound Heights
Let the original height from which the ball is dropped be \( H \). When the ball bounces, it reaches a height \( h' \) which is 80% of \( H \). Therefore, we can express \( h' \) as:
\[ h' = 0.8H \]

### Step 2: Calculate the Potential Energy at the Original Height
The potential energy (PE) of the ball at the original height \( H \) is given by the formula:
\[ PE = mgh \]
Substituting \( H \) into the formula, we have:
\[ PE_{initial} = mgH \]

### Step 3: Calculate the Potential Energy at the Rebound Height
The potential energy of the ball at the rebound height \( h' \) is:
\[ PE_{rebound} = mg(h') = mg(0.8H) \]
This simplifies to:
\[ PE_{rebound} = 0.8mgH \]

### Step 4: Determine the Change in Potential Energy
The change in potential energy (which corresponds to the mechanical energy lost) when the ball bounces is given by:
\[ \Delta PE = PE_{initial} - PE_{rebound} \]
Substituting the values we calculated:
\[ \Delta PE = mgH - 0.8mgH = mgH(1 - 0.8) = mgH(0.2) \]

### Step 5: Calculate the Fraction of Mechanical Energy Lost
To find the fraction of mechanical energy lost, we can use the formula:
\[ \text{Fractional Loss} = \frac{\Delta PE}{PE_{initial}} \]
Substituting the values we have:
\[ \text{Fractional Loss} = \frac{0.2mgH}{mgH} = 0.2 \]

### Conclusion
Thus, the fraction of mechanical energy lost in each bounce is:
\[ \text{Fraction of energy lost} = 0.2 \]