A ball dropped from height h on a horizontal floor goes up to the height `(3h)/(4)` after hitting the floor. Fraction of energy of ball lost in the impact is

Options

A. `(3)/(4)`
B. `(1)/(2)`
C. `(3)/(5)`
D. `(1)/(4)`

Answer

To find the fraction of energy lost by the ball after it hits the floor, we can follow these steps:

### Step 1: Calculate the Initial Potential Energy (Ei)
When the ball is dropped from a height \( h \), its initial potential energy (Ei) can be calculated using the formula:
\[ E_i = mgh \]
where \( m \) is the mass of the ball, \( g \) is the acceleration due to gravity, and \( h \) is the height from which the ball is dropped.

### Step 2: Calculate the Final Potential Energy (Ef)
After hitting the floor, the ball rises to a height of \( \frac{3h}{4} \). The final potential energy (Ef) can be calculated as:
\[ E_f = mg \left( \frac{3h}{4} \right) = \frac{3}{4} mgh \]

### Step 3: Calculate the Change in Energy (ΔE)
The change in energy (ΔE) is the difference between the initial potential energy and the final potential energy:
\[ \Delta E = E_i - E_f \]
Substituting the values we calculated:
\[ \Delta E = mgh - \frac{3}{4} mgh \]
Factoring out \( mgh \):
\[ \Delta E = mgh \left( 1 - \frac{3}{4} \right) = mgh \left( \frac{1}{4} \right) \]

### Step 4: Calculate the Fraction of Energy Lost
The fraction of energy lost can be calculated using the formula:
\[ \text{Fraction of energy lost} = \frac{\Delta E}{E_i} \]
Substituting the values we have:
\[ \text{Fraction of energy lost} = \frac{\frac{1}{4} mgh}{mgh} \]
The \( mgh \) terms cancel out:
\[ \text{Fraction of energy lost} = \frac{1}{4} \]

### Conclusion
The fraction of energy lost by the ball after the impact is \( \frac{1}{4} \).

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