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 1: Calculate the velocity just before the collision
The ball is dropped from a height of 5 m. We can use the equation of motion to find the velocity just before it hits the ground.

Using the formula:
\[ v^2 = u^2 + 2gh \]
where:
- \( u = 0 \) (initial velocity, since the ball is dropped)
- \( g = 10 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( h = 5 \, \text{m} \) (height from which the ball is dropped)

Substituting the values:
\[ v^2 = 0 + 2 \times 10 \times 5 \]
\[ v^2 = 100 \]
\[ v = \sqrt{100} = 10 \, \text{m/s} \]

### Step 2: Calculate the velocity just after the collision
After bouncing back to a height of 1.8 m, we need to find the velocity just after the collision. Again, we can use the same equation of motion.

Using the formula:
\[ v^2 = u^2 + 2gh \]
where:
- \( u \) is the initial velocity just after the collision (which we need to find)
- \( g = 10 \, \text{m/s}^2 \)
- \( h = 1.8 \, \text{m} \)

At the maximum height after the bounce, the final velocity \( v = 0 \):
\[ 0 = u^2 - 2 \times 10 \times 1.8 \]
\[ u^2 = 2 \times 10 \times 1.8 \]
\[ u^2 = 36 \]
\[ u = \sqrt{36} = 6 \, \text{m/s} \]

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

Using the formula:
\[ e = \frac{\text{Velocity after collision}}{\text{Velocity before collision}} \]

Before the collision, the velocity of the ball is \( 10 \, \text{m/s} \) (downward), and after the collision, the velocity of the ball is \( 6 \, \text{m/s} \) (upward).

Thus:
\[ e = \frac{6}{10} = 0.6 \]

### Final Answer
The coefficient of restitution for the collision is \( 0.6 \).

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