A ball of mass m, travelling with velocity `2hati+3hatj` receives an impulse `-3mhati`. What is the velocity of the ball immediately afterwards?

Explanation

To solve the problem, we need to find the velocity of a ball after it receives an impulse. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Identify Given Values**:
- Initial velocity of the ball, \( \vec{v_i} = 2\hat{i} + 3\hat{j} \)
- Impulse received, \( \vec{J} = -3m\hat{i} \)
- Mass of the ball, \( m \)

2. **Understand the Concept of Impulse**:
- Impulse is defined as the change in momentum. Mathematically, it can be expressed as:
\[ \vec{J} = \Delta \vec{p} = m\vec{v_f} - m\vec{v_i} \]
- Here, \( \vec{v_f} \) is the final velocity we want to find.

3. **Set Up the Equation**:
- Rearranging the impulse-momentum equation gives:
\[ \vec{J} = m\vec{v_f} - m\vec{v_i} \]
- This can be rewritten as:
\[ \vec{J} + m\vec{v_i} = m\vec{v_f} \]

4. **Substituting the Values**:
- Substitute the known values into the equation:
\[ -3m\hat{i} + m(2\hat{i} + 3\hat{j}) = m\vec{v_f} \]

5. **Simplifying the Equation**:
- Distributing \( m \) in the equation:
\[ -3m\hat{i} + 2m\hat{i} + 3m\hat{j} = m\vec{v_f} \]
- Combine the terms:
\[ (-3m + 2m)\hat{i} + 3m\hat{j} = m\vec{v_f} \]
- This simplifies to:
\[ -m\hat{i} + 3m\hat{j} = m\vec{v_f} \]

6. **Dividing by Mass**:
- Since \( m \) is non-zero, we can divide through by \( m \):
\[ \vec{v_f} = -\hat{i} + 3\hat{j} \]

7. **Final Result**:
- Therefore, the final velocity of the ball immediately after receiving the impulse is:
\[ \vec{v_f} = -\hat{i} + 3\hat{j} \]

### Final Answer:
The velocity of the ball immediately afterwards is \( -\hat{i} + 3\hat{j} \).

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