A 1.0 kg ball drops vertically into a floor from a height of 25 cm . It rebounds to a height of 4 cm. The coefficient of restitution for the collision is

Options

A. 0.16
B. 0.32
C. 0.4
D. 0.56

Answer

To find the coefficient of restitution (e) for the collision of the ball with the floor, we will follow these steps:

### Step 1: Determine the velocity just before impact
When the ball is dropped from a height of 25 cm, we can calculate the velocity just before it hits the ground using the formula for free fall:

\[ v = \sqrt{2gh} \]

Where:
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \))
- \( h \) is the height from which the ball is dropped (25 cm = 0.25 m)

Substituting the values:

\[ v = \sqrt{2 \times 9.8 \, \text{m/s}^2 \times 0.25 \, \text{m}} = \sqrt{4.9} \approx 2.21 \, \text{m/s} \]

### Step 2: Determine the velocity just after impact
After rebounding to a height of 4 cm, we can find the velocity just after the impact using the same formula:

\[ v' = \sqrt{2gh'} \]

Where \( h' \) is the height to which the ball rebounds (4 cm = 0.04 m):

\[ v' = \sqrt{2 \times 9.8 \, \text{m/s}^2 \times 0.04 \, \text{m}} = \sqrt{0.784} \approx 0.88 \, \text{m/s} \]

### Step 3: Calculate the coefficient of restitution
The coefficient of restitution (e) is defined as the ratio of the velocity after the collision to the velocity before the collision:

\[ e = \frac{v'}{v} \]

Substituting the values we calculated:

\[ e = \frac{0.88 \, \text{m/s}}{2.21 \, \text{m/s}} \approx 0.398 \approx 0.4 \]

### Final Answer
Thus, the coefficient of restitution for the collision is approximately **0.4**.

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