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. (a) `1/2s`
B. (b) `1s`
C. (c) `3/2s`
D. (d) `3/4s`

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the velocity just before the first impact
The ball falls freely from a height of \( h = 4.9 \, \text{m} \). We can use the formula for the velocity of an object in free fall:

\[ v = \sqrt{2gh} \]

Where:
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( h = 4.9 \, \text{m} \)

Substituting the values:

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

### Step 2: Apply the coefficient of restitution
The coefficient of restitution \( e \) is given as \( \frac{3}{4} \). The velocity of the ball just after the collision can be calculated using the formula:

\[ v' = e \times v \]

Where:
- \( v' \) is the velocity after the collision
- \( e = \frac{3}{4} \)
- \( v = 9.8 \, \text{m/s} \)

Substituting the values:

\[ v' = \frac{3}{4} \times 9.8 = 7.35 \, \text{m/s} \]

### Step 3: Calculate the time taken to reach the ground after the first bounce
The time taken to reach the ground after the first bounce can be calculated using the formula:

\[ t = \frac{2u}{g} \]

Where:
- \( u = v' = 7.35 \, \text{m/s} \)
- \( g = 9.8 \, \text{m/s}^2 \)

Substituting the values:

\[ t = \frac{2 \times 7.35}{9.8} = \frac{14.7}{9.8} \approx 1.5 \, \text{s} \]

### Step 4: Calculate the total time until the second strike
The total time until the second strike is the sum of the time taken to fall the first distance and the time taken to fall the second distance. The time taken to fall the first distance is the time taken to fall from the height of \( 4.9 \, \text{m} \), which we calculated as:

\[ t_1 = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 4.9}{9.8}} = 1 \, \text{s} \]

Now, the total time until the second strike is:

\[ t_{\text{total}} = t_1 + t_2 = 1 \, \text{s} + 1.5 \, \text{s} = 2.5 \, \text{s} \]

### Final Answer
The ball will strike the ground for the second time after approximately \( 2.5 \, \text{s} \).

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