A force exerts an impulse Ion a particle changing its speed from initial velocity `u` to final velocity `2u`. The applied force and the initial velocity are oppositely oriented along the same line. The work done by the force is

Options

A. `3/2Iu`
B. `1/2Iu`
C. `Iu`
D. `2Iu`

Explanation

To solve the problem, we need to find the work done by the force when it changes the speed of a particle from an initial velocity \( u \) to a final velocity \( 2u \). The force is applied in the opposite direction to the initial velocity.

### Step-by-Step Solution:

1. **Understanding the Impulse**:
The impulse \( I \) applied to the particle is given by the change in momentum. The initial momentum \( p_i \) is \( m \cdot u \) (where \( m \) is the mass of the particle), and the final momentum \( p_f \) is \( m \cdot (2u) \). Since the force is applied in the opposite direction, we can express the impulse as:
\[ I = p_f - p_i = m \cdot (2u) - m \cdot u = m \cdot (2u - u) = m \cdot u \]

2. **Calculating the Change in Kinetic Energy**:
The initial kinetic energy \( KE_i \) is given by:
\[ KE_i = \frac{1}{2} m u^2 \]
The final kinetic energy \( KE_f \) is given by:
\[ KE_f = \frac{1}{2} m (2u)^2 = \frac{1}{2} m \cdot 4u^2 = 2 m u^2 \]
The change in kinetic energy \( \Delta KE \) is:
\[ \Delta KE = KE_f - KE_i = 2mu^2 - \frac{1}{2} mu^2 = \frac{4mu^2}{2} - \frac{1}{2} mu^2 = \frac{3}{2} mu^2 \]

3. **Relating Work Done to Change in Kinetic Energy**:
According to the work-energy theorem, the work done \( W \) by the force is equal to the change in kinetic energy:
\[ W = \Delta KE = \frac{3}{2} mu^2 \]

4. **Expressing Work Done in Terms of Impulse**:
We previously found that the impulse \( I \) is equal to \( mu \). Since the impulse is related to the change in momentum and we have:
\[ I = 3mu \]
We can express \( mu \) in terms of impulse:
\[ mu = \frac{I}{3} \]
Substituting this into the expression for work done:
\[ W = \frac{3}{2} m u^2 = \frac{3}{2} \left(\frac{I}{3}\right) u = \frac{Iu}{2} \]

5. **Final Expression**:
Therefore, the work done by the force is:
\[ W = \frac{Iu}{2} \]

### Conclusion:
The work done by the force is \( \frac{Iu}{2} \).