To solve the problem of finding the impulse imparted to the ball when a batsman deflects it by an angle of 60° without changing its initial speed of 20 m/s, 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 \( p_i \) can be calculated using the formula:
\[ p_i = m \cdot v_i \]
where:
- \( m = 0.15 \, \text{kg} \) (mass of the ball)
- \( v_i = 20 \, \text{m/s} \) (initial speed)
Thus,
\[ p_i = 0.15 \, \text{kg} \cdot 20 \, \text{m/s} = 3 \, \text{kg m/s} \]
### Step 3: Determine the final velocity components
After deflecting the ball by an angle of 60°, the components of the final velocity \( v_f \) can be expressed as:
- The x-component: \( v_{fx} = v_i \cdot \cos(60°) \)
- The y-component: \( v_{fy} = v_i \cdot \sin(60°) \)
Calculating these components:
- \( \cos(60°) = 0.5 \)
- \( \sin(60°) = \frac{\sqrt{3}}{2} \)
Thus,
\[ v_{fx} = 20 \cdot 0.5 = 10 \, \text{m/s} \]
\[ v_{fy} = 20 \cdot \frac{\sqrt{3}}{2} = 10\sqrt{3} \, \text{m/s} \]
### Step 4: Calculate the final momentum
The final momentum \( p_f \) can be calculated as:
\[ p_f = m \cdot v_f \]
where \( v_f \) is the velocity vector:
\[ v_f = (10 \, \text{m/s} \, \hat{i} + 10\sqrt{3} \, \text{m/s} \, \hat{j}) \]
Thus,
\[ p_f = 0.15 \cdot (10 \hat{i} + 10\sqrt{3} \hat{j}) \]
\[ p_f = 1.5 \hat{i} + 1.5\sqrt{3} \hat{j} \, \text{kg m/s} \]
### Step 5: Calculate the change in momentum (impulse)
Now, we can find the impulse:
\[ \text{Impulse} = p_f - p_i \]
\[ p_i = 3 \hat{i} + 0 \hat{j} \]
So,
\[ \text{Impulse} = (1.5 \hat{i} + 1.5\sqrt{3} \hat{j}) - (3 \hat{i}) \]
\[ \text{Impulse} = (-1.5 \hat{i} + 1.5\sqrt{3} \hat{j}) \]
### Step 6: Calculate the magnitude of the impulse
The magnitude of the impulse can be calculated using the Pythagorean theorem:
\[ |\text{Impulse}| = \sqrt{(-1.5)^2 + (1.5\sqrt{3})^2} \]
Calculating this gives:
\[ |\text{Impulse}| = \sqrt{2.25 + 6.75} = \sqrt{9} = 3 \, \text{kg m/s} \]
### Final Answer
The impulse imparted to the ball is \( 3 \, \text{kg m/s} \).
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