A metal ball falls from a height of 1m on to a steel plate and jumps upto a height of 81 cm. The coefficient of restitution of the ball and steel plate is

Options

A. 0.2
B. 9
C. 0.9
D. 90

Answer

To find the coefficient of restitution (e) between the metal ball and the steel plate, we can follow these steps:

### Step 1: Understand the Problem
The metal ball falls from a height of 1 meter (h1) and bounces back to a height of 81 cm (h2). We need to find the coefficient of restitution, which is defined as the ratio of the velocity of separation to the velocity of approach.

### Step 2: Use the Formula for Velocity
The velocity of the ball just before it hits the steel plate (v1) can be calculated using the formula for free fall:
\[ v_1 = \sqrt{2gh_1} \]
Where:
- \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \))
- \( h_1 = 1 \, \text{m} \)

### Step 3: Calculate v1
Substituting the values:
\[ v_1 = \sqrt{2 \cdot 9.81 \cdot 1} \]
\[ v_1 = \sqrt{19.62} \]
\[ v_1 \approx 4.43 \, \text{m/s} \]

### Step 4: Calculate the Velocity After the Bounce
The velocity of the ball just after it bounces back to a height of 81 cm (h2) can be calculated using the same formula:
\[ v_2 = \sqrt{2gh_2} \]
Where:
- \( h_2 = 81 \, \text{cm} = 0.81 \, \text{m} \)

### Step 5: Calculate v2
Substituting the values:
\[ v_2 = \sqrt{2 \cdot 9.81 \cdot 0.81} \]
\[ v_2 = \sqrt{15.8762} \]
\[ v_2 \approx 3.98 \, \text{m/s} \]

### Step 6: Find the Coefficient of Restitution
The coefficient of restitution (e) is given by:
\[ e = \frac{v_2}{v_1} \]

### Step 7: Substitute the Values
Substituting the calculated values:
\[ e = \frac{3.98}{4.43} \]
\[ e \approx 0.9 \]

### Final Answer
The coefficient of restitution of the ball and steel plate is approximately **0.9**.

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