A body falling from a height of 10m rebounds from hard floor. If it loses 20% energy in the impact, then coefficient of restitution is

Options

A. 0.89
B. 0.56
C. 0.23
D. 0.18

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Calculate the velocity just before impact
When a body falls from a height \( h \), its velocity just before impact can be calculated using the equation of motion:
\[ v = \sqrt{2gh} \]
Where:
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h \) is the height (10 m in this case)

Substituting the values:
\[ v = \sqrt{2 \times 9.81 \times 10} = \sqrt{196.2} \approx 14.0 \, \text{m/s} \]

### Step 2: Calculate the initial kinetic energy
The initial kinetic energy (KE_initial) just before the impact can be calculated using:
\[ KE_{\text{initial}} = \frac{1}{2} mv^2 \]
Substituting \( v \):
\[ KE_{\text{initial}} = \frac{1}{2} m (14.0)^2 = \frac{1}{2} m \times 196 \approx 98m \, \text{J} \]

### Step 3: Determine the energy lost during impact
It is given that the body loses 20% of its energy during the impact. Therefore, the energy lost (E_lost) is:
\[ E_{\text{lost}} = 0.2 \times KE_{\text{initial}} = 0.2 \times 98m = 19.6m \, \text{J} \]

### Step 4: Calculate the final kinetic energy after impact
The final kinetic energy (KE_final) after the impact can be calculated as:
\[ KE_{\text{final}} = KE_{\text{initial}} - E_{\text{lost}} = 98m - 19.6m = 78.4m \, \text{J} \]

### Step 5: Relate final kinetic energy to final velocity
Using the final kinetic energy to find the final velocity (\( v' \)):
\[ KE_{\text{final}} = \frac{1}{2} mv'^2 \]
Substituting for KE_final:
\[ 78.4m = \frac{1}{2} mv'^2 \]
Canceling \( m \) from both sides:
\[ 78.4 = \frac{1}{2} v'^2 \implies v'^2 = 156.8 \implies v' = \sqrt{156.8} \approx 12.5 \, \text{m/s} \]

### Step 6: Calculate the coefficient of restitution
The coefficient of restitution (e) is defined as the ratio of the velocity of separation to the velocity of approach:
\[ e = \frac{v'}{v} \]
Substituting the values of \( v' \) and \( v \):
\[ e = \frac{12.5}{14.0} \approx 0.893 \]

### Final Answer
The coefficient of restitution is approximately \( 0.89 \).

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