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. `1/2s`
B. `1s`
C. `3/2s`
D. `3/4s`

Explanation

To solve the problem step by step, we will analyze the motion of the ball as it falls, rebounds, and then falls again.

### Step 1: Calculate the velocity just before the first impact
The ball is dropped from a height of \( h = 4.9 \, \text{m} \). We can use the equation of motion to find the velocity just before it strikes the ground.

Using the formula:
\[ v = \sqrt{2gh} \]
where \( g = 9.8 \, \text{m/s}^2 \).

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

### Step 2: Calculate the velocity after the first impact
The coefficient of restitution \( e \) is given as \( \frac{3}{4} \). The velocity after the impact can be calculated using the formula:
\[ v_f = e \cdot v \]
Substituting the values:
\[ v_f = \frac{3}{4} \times 9.8 \approx 7.35 \, \text{m/s} \]

### Step 3: Calculate the time taken to reach the maximum height after the first impact
When the ball rebounds, it will rise to a certain height before coming to rest momentarily. We can find the time taken to reach this maximum height using the formula:
\[ t = \frac{v_f}{g} \]
Substituting the values:
\[ t = \frac{7.35}{9.8} \approx 0.75 \, \text{s} \]

### Step 4: Calculate the height reached after the first impact
We can find the height reached using the formula:
\[ h' = \frac{v_f^2}{2g} \]
Substituting the values:
\[ h' = \frac{(7.35)^2}{2 \times 9.8} \approx \frac{54.0225}{19.6} \approx 2.75 \, \text{m} \]

### Step 5: Calculate the time taken to fall back to the ground from the maximum height
The time taken to fall from the maximum height back to the ground can be calculated using the formula:
\[ t' = \sqrt{\frac{2h'}{g}} \]
Substituting the values:
\[ t' = \sqrt{\frac{2 \times 2.75}{9.8}} \approx \sqrt{\frac{5.5}{9.8}} \approx \sqrt{0.5612} \approx 0.75 \, \text{s} \]

### Step 6: Calculate the total time between the first and second impact
The total time between the first and second impact is the sum of the time taken to rise and the time taken to fall:
\[ T = t + t' = 0.75 + 0.75 = 1.5 \, \text{s} \]

### Final Answer
The ball will strike the ground for the second time after \( 1.5 \, \text{s} \).

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