A ball is dropped on the floor from a height of 10m. It rebounds to a height of 2.5 m if the ball is in contact with floor for 0.01 s then the average acceleration during contact is nearly

Options

A. `500sqrt(2) m//s^(2)` upwards
B. `1800sqrt(2) m//s^(2)` downwards
C. `1500sqrt(2) m//s^(2)` upwards
D. `1500sqrt(2) m//s^(2)` downwards

Explanation

To solve the problem of finding the average acceleration of the ball during its contact with the floor, we can follow these steps:

### Step 1: Calculate the velocity just before impact (V1)
The ball is dropped from a height of 10 m. We can use the equation of motion to find the velocity just before it hits the floor.

Using the formula:
\[ V^2 = U^2 + 2gh \]
Where:
- \( V \) = final velocity (just before impact)
- \( U \) = initial velocity (0 m/s, since the ball is dropped)
- \( g \) = acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \))
- \( h \) = height (10 m)

Substituting the values:
\[ V^2 = 0 + 2 \cdot 10 \cdot 10 \]
\[ V^2 = 200 \]
\[ V = \sqrt{200} = 10\sqrt{2} \, \text{m/s} \]

### Step 2: Calculate the velocity just after rebound (V2)
The ball rebounds to a height of 2.5 m. We can again use the equation of motion to find the velocity just after it leaves the floor.

Using the same formula:
\[ V^2 = U^2 + 2gh \]
Where:
- \( V \) = final velocity (0 m/s at the peak of the rebound)
- \( U \) = initial velocity (which we need to find)
- \( g \) = acceleration due to gravity (10 m/s²)
- \( h \) = height (2.5 m)

Rearranging the formula:
\[ 0 = U^2 - 2 \cdot 10 \cdot 2.5 \]
\[ U^2 = 50 \]
\[ U = \sqrt{50} = 5\sqrt{2} \, \text{m/s} \]

### Step 3: Calculate the change in velocity (ΔV)
The change in velocity during the contact with the floor can be calculated as:
\[ \Delta V = V2 - V1 \]
Where:
- \( V2 = 5\sqrt{2} \, \text{m/s} \) (upward)
- \( V1 = -10\sqrt{2} \, \text{m/s} \) (downward, hence negative)

Substituting the values:
\[ \Delta V = 5\sqrt{2} - (-10\sqrt{2}) \]
\[ \Delta V = 5\sqrt{2} + 10\sqrt{2} \]
\[ \Delta V = 15\sqrt{2} \, \text{m/s} \]

### Step 4: Calculate the average acceleration (a_avg)
The average acceleration can be calculated using the formula:
\[ a_{avg} = \frac{\Delta V}{\Delta t} \]
Where:
- \( \Delta t = 0.01 \, \text{s} \)

Substituting the values:
\[ a_{avg} = \frac{15\sqrt{2}}{0.01} \]
\[ a_{avg} = 1500\sqrt{2} \, \text{m/s}^2 \]

### Final Answer
The average acceleration during contact is approximately \( 1500\sqrt{2} \, \text{m/s}^2 \).

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