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. (a)0.16
B. (b)0.32
C. (c)0.4
D. (d)0.56

Answer

To find the coefficient of restitution for the collision of a ball dropped from a height of 25 cm and rebounding to a height of 4 cm, we can follow these steps:

### Step 1: Understand the Coefficient of Restitution
The coefficient of restitution (e) is defined as the ratio of the relative speed after the collision to the relative speed before the collision. It can also be expressed in terms of the heights before and after the collision:
\[ e = \sqrt{\frac{h'}{h}} \]
where \( h' \) is the rebound height and \( h \) is the drop height.

### Step 2: Identify the Heights
From the problem:
- The height from which the ball is dropped, \( h = 25 \) cm = \( 0.25 \) m.
- The height to which the ball rebounds, \( h' = 4 \) cm = \( 0.04 \) m.

### Step 3: Substitute the Heights into the Coefficient of Restitution Formula
Using the formula for the coefficient of restitution:
\[ e = \sqrt{\frac{h'}{h}} = \sqrt{\frac{0.04}{0.25}} \]

### Step 4: Calculate the Ratio of Heights
Now calculate the ratio:
\[ \frac{0.04}{0.25} = \frac{4}{25} \]

### Step 5: Take the Square Root
Now take the square root:
\[ e = \sqrt{\frac{4}{25}} = \frac{2}{5} = 0.4 \]

### Step 6: Conclusion
The coefficient of restitution for the collision is:
\[ e = 0.4 \]

### Summary
Thus, the coefficient of restitution for the collision of the ball is **0.4**.

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