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

Answer

To solve the problem, we need to find the impulse given to a particle of mass \( m \) that changes its velocity from \( v_1 \) to \( v_2 \).

### Step-by-Step Solution:

1. **Understand the Concept of Impulse**:
Impulse is defined as the change in momentum of an object. Mathematically, impulse \( I \) can be expressed as:
\[ I = \Delta P \]
where \( \Delta P \) is the change in momentum.

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

3. **Calculate Initial and Final Momentum**:
- Initial momentum \( P_1 \) when the particle is moving with velocity \( v_1 \):
\[ P_1 = m \cdot v_1 \]
- Final momentum \( P_2 \) when the velocity becomes \( v_2 \):
\[ P_2 = m \cdot v_2 \]

4. **Find the Change in Momentum**:
The change in momentum \( \Delta P \) is given by:
\[ \Delta P = P_2 - P_1 = m \cdot v_2 - m \cdot v_1 \]
Simplifying this, we get:
\[ \Delta P = m(v_2 - v_1) \]

5. **Express Impulse in Terms of Change in Velocity**:
Since impulse \( I \) is equal to the change in momentum, we can write:
\[ I = \Delta P = m(v_2 - v_1) \]

### Final Answer:
Thus, the impulse given to the particle is:
\[ I = m(v_2 - v_1) \]