A ball of mass 400 gm is dropped from a height of 5 m. A boy on the ground hits the ball vertically upwards with a bat with an average force of 100 newton so that it attains a vertical velocity of 20 m/s. The time for which the ball remains in contact with the bat is `(g=10m//s^(2))`

Options

A. 0.12 s
B. 0.08s
C. 0.04 s
D. 12s

Explanation

To solve the problem step by step, we will follow the physics principles involved, particularly focusing on the concepts of momentum and forces.

### Step 1: Calculate the velocity of the ball just before it hits the ground.
The ball is dropped from a height of 5 m. We can use the equation of motion:

\[ v^2 = u^2 + 2gh \]

Where:
- \( v \) = final velocity
- \( u \) = initial velocity (0 m/s, since it is dropped)
- \( g \) = acceleration due to gravity (10 m/s²)
- \( h \) = height (5 m)

Substituting the values:

\[ v^2 = 0 + 2 \times 10 \times 5 \]
\[ v^2 = 100 \]
\[ v = \sqrt{100} \]
\[ v = 10 \, \text{m/s} \]

### Step 2: Determine the change in momentum when the ball is hit by the bat.
The initial momentum of the ball just before it hits the bat is:

\[ p_{initial} = m \cdot v \]

Where:
- \( m \) = mass of the ball = 400 g = 0.4 kg
- \( v \) = 10 m/s (calculated above)

Calculating the initial momentum:

\[ p_{initial} = 0.4 \cdot 10 = 4 \, \text{kg m/s} \]

The final momentum of the ball after being hit by the bat (with a final velocity of 20 m/s upwards) is:

\[ p_{final} = m \cdot v_{final} = 0.4 \cdot 20 = 8 \, \text{kg m/s} \]

### Step 3: Calculate the change in momentum.
The change in momentum (\( \Delta p \)) is given by:

\[ \Delta p = p_{final} - p_{initial} \]

Substituting the values:

\[ \Delta p = 8 - 4 = 4 \, \text{kg m/s} \]

### Step 4: Use Newton's second law to find the time of contact.
According to Newton's second law, the force is equal to the rate of change of momentum:

\[ F = \frac{\Delta p}{\Delta t} \]

Rearranging to find \( \Delta t \):

\[ \Delta t = \frac{\Delta p}{F} \]

Substituting the values:

\[ \Delta t = \frac{4 \, \text{kg m/s}}{100 \, \text{N}} \]

Calculating \( \Delta t \):

\[ \Delta t = 0.04 \, \text{s} \]

### Final Answer:
The time for which the ball remains in contact with the bat is **0.04 seconds**.

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