A rubber ball is dropped from a height of 5 m on a plane. If bounces back to a height of 1.8 m. Find the coefficient of the resitution for the collision.

Answer

To find the coefficient of restitution for the collision of a rubber ball dropped from a height of 5 m and bouncing back to a height of 1.8 m, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Coefficient of Restitution (e)**:
The coefficient of restitution (e) is defined as the ratio of the relative velocity of separation to the relative velocity of approach between two colliding bodies. In this case, we can relate it to the heights before and after the bounce.

2. **Identify the Heights**:
- Initial height (h) = 5 m (the height from which the ball is dropped)
- Rebound height (h') = 1.8 m (the height to which the ball bounces back)

3. **Use the Formula for Coefficient of Restitution**:
The relationship between the heights and the coefficient of restitution is given by:
\[ e = \sqrt{\frac{h'}{h}} \]

4. **Substitute the Values**:
Substitute the values of h and h' into the formula:
\[ e = \sqrt{\frac{1.8}{5}} \]

5. **Calculate the Ratio**:
Calculate the ratio:
\[ \frac{1.8}{5} = 0.36 \]

6. **Take the Square Root**:
Now, take the square root of 0.36 to find e:
\[ e = \sqrt{0.36} = 0.6 \]

7. **Final Answer**:
Therefore, the coefficient of restitution for the collision is:
\[ e = 0.6 \]