A ball after freely falling from a height of `4.9m` strikes a horizontal plane. If the coefficient of restitution is `3/4`, the ball will strike second time with the plane after

Options

A. 1s
B. 1.5 s
C. 2 s
D. 3 s

Explanation

To solve the problem step by step, we will follow the given information and apply the relevant physics concepts.

### Step 1: Identify the given values
- Height (h) = 4.9 m
- Coefficient of restitution (e) = 3/4

### Step 2: Calculate the velocity just before the first impact
The velocity of the ball just before it strikes the ground can be calculated using the formula for free fall:
\[ v = \sqrt{2gh} \]
Where:
- \( g \) = acceleration due to gravity = 9.8 m/s²
- \( h \) = height = 4.9 m

Substituting the values:
\[ v = \sqrt{2 \times 9.8 \times 4.9} \]
\[ v = \sqrt{96.04} \]
\[ v \approx 9.8 \, \text{m/s} \]

### Step 3: Calculate the velocity after the first impact
Using the coefficient of restitution, the velocity after the collision (u) can be calculated as:
\[ u = e \times v \]
Where:
- \( e \) = coefficient of restitution = 3/4
- \( v \) = velocity before impact = 9.8 m/s

Substituting the values:
\[ u = \frac{3}{4} \times 9.8 \]
\[ u = 7.35 \, \text{m/s} \]

### Step 4: Calculate the time taken to reach the ground after the first bounce
The time taken to reach the maximum height after the first bounce can be calculated using the formula:
\[ t = \frac{u}{g} \]
Where:
- \( g \) = acceleration due to gravity = 9.8 m/s²

Substituting the values:
\[ t = \frac{7.35}{9.8} \]
\[ t \approx 0.75 \, \text{s} \]

### Step 5: Calculate the time taken to fall back to the ground
The time taken to fall back to the ground from the maximum height is the same as the time taken to rise:
\[ t_{fall} = t = 0.75 \, \text{s} \]

### Step 6: Calculate the total time for the second strike
The total time until the ball strikes the ground for the second time is the sum of the time taken to rise and the time taken to fall:
\[ T = t + t_{fall} \]
\[ T = 0.75 + 0.75 \]
\[ T = 1.5 \, \text{s} \]

### Final Answer
The ball will strike the ground for the second time after **1.5 seconds**.

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