A particle of mass m is moving with a uniform velocity `v_(1)`. It is given an impulse such that its velocity becomes `v_(2)`. The impulse is equal to

Options

A. `m(v_(1)+v_(2))`
B. `m[|v_(2)|-|v_(1)|]`
C. `1/2(v_(2)^(2)-v_(1)^(2))`
D. `m(v_(2)-v_(1))`

Explanation

To solve the problem of finding the impulse imparted to a particle of mass \( m \) that changes its velocity from \( v_1 \) to \( v_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Impulse**:
Impulse is defined as the change in momentum of an object. Mathematically, it can be expressed as:
\[ \text{Impulse} = \Delta p = p_{\text{final}} - p_{\text{initial}} \]
where \( p \) is the momentum of the object.

2. **Define Momentum**:
The momentum \( p \) of an object is given by the product of its mass \( m \) and its velocity \( v \):
\[ p = m \cdot v \]

3. **Calculate Initial Momentum**:
For the particle moving with an initial velocity \( v_1 \), the initial momentum \( p_{\text{initial}} \) is:
\[ p_{\text{initial}} = m \cdot v_1 \]

4. **Calculate Final Momentum**:
After the impulse is applied, the particle moves with a new velocity \( v_2 \). Thus, the final momentum \( p_{\text{final}} \) is:
\[ p_{\text{final}} = m \cdot v_2 \]

5. **Find the Change in Momentum**:
The change in momentum \( \Delta p \) can now be calculated as:
\[ \Delta p = p_{\text{final}} - p_{\text{initial}} = (m \cdot v_2) - (m \cdot v_1) \]

6. **Factor Out the Mass**:
We can factor out the mass \( m \) from the equation:
\[ \Delta p = m(v_2 - v_1) \]

7. **Conclusion**:
Therefore, the impulse imparted to the particle is:
\[ \text{Impulse} = m(v_2 - v_1) \]

### Final Answer:
The impulse is equal to \( m(v_2 - v_1) \).

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