A batsman hits back a ball straight in the direction of the bowler without changing its initial speed of `12 ms^(-1)` . If the mass of the ball is 0.15 kg , determine the impulse imparted to the ball . (Assume linear motion of the ball).

Options

A. 1.8N s
B. 2.8N s
C. 3.6N s
D. 4.2N s

Explanation

To determine the impulse imparted to the ball, we can follow these steps:

### Step 1: Understand the concept of impulse
Impulse is defined as the change in momentum of an object. Mathematically, it can be expressed as:
\[ \text{Impulse} = \Delta p = p_f - p_i \]
where \( p_f \) is the final momentum and \( p_i \) is the initial momentum.

### Step 2: Calculate the initial momentum
The initial momentum \( p_i \) of the ball can be calculated using the formula:
\[ p_i = m \cdot v_i \]
where:
- \( m \) is the mass of the ball (0.15 kg)
- \( v_i \) is the initial velocity of the ball (12 m/s)

So,
\[ p_i = 0.15 \, \text{kg} \cdot 12 \, \text{m/s} = 1.8 \, \text{kg m/s} \]

### Step 3: Calculate the final momentum
When the batsman hits the ball back, the direction of the ball changes. The final velocity \( v_f \) of the ball will be -12 m/s (negative because it is in the opposite direction). Thus, the final momentum \( p_f \) is:
\[ p_f = m \cdot v_f \]
So,
\[ p_f = 0.15 \, \text{kg} \cdot (-12) \, \text{m/s} = -1.8 \, \text{kg m/s} \]

### Step 4: Calculate the change in momentum
Now, we can find the change in momentum:
\[ \Delta p = p_f - p_i \]
Substituting the values we calculated:
\[ \Delta p = -1.8 \, \text{kg m/s} - 1.8 \, \text{kg m/s} = -3.6 \, \text{kg m/s} \]

### Step 5: Determine the impulse
The impulse imparted to the ball is equal to the change in momentum:
\[ \text{Impulse} = -3.6 \, \text{kg m/s} \]

Since impulse is often expressed in terms of magnitude, we can say:
\[ \text{Impulse} = 3.6 \, \text{N s} \]

### Final Answer
The impulse imparted to the ball is **3.6 N s**.

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