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

Options

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

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a particle that changes its speed from \( u \) to \( 2u \) under the influence of a force that is acting in the opposite direction to its initial velocity. We need to find the work done by this force in terms of the impulse \( I \).

### Step 2: Use the Work-Energy Theorem
According to the work-energy theorem, the work done \( W \) by the force is equal to the change in kinetic energy of the particle:
\[ W = K_f - K_i \]
where \( K_f \) is the final kinetic energy and \( K_i \) is the initial kinetic energy.

### Step 3: Calculate Initial and Final Kinetic Energies
The initial kinetic energy \( K_i \) when the speed is \( 2u \) is given by:
\[ K_i = \frac{1}{2} m (2u)^2 = \frac{1}{2} m (4u^2) = 2mu^2 \]

The final kinetic energy \( K_f \) when the speed is \( u \) is given by:
\[ K_f = \frac{1}{2} m (u)^2 = \frac{1}{2} mu^2 \]

### Step 4: Calculate the Change in Kinetic Energy
Now, we can find the change in kinetic energy:
\[ \Delta K = K_f - K_i = \frac{1}{2} mu^2 - 2mu^2 = \frac{1}{2} mu^2 - \frac{4}{2} mu^2 = -\frac{3}{2} mu^2 \]

### Step 5: Substitute into the Work Equation
Thus, the work done by the force is:
\[ W = -\frac{3}{2} mu^2 \]

### Step 6: Relate Impulse to Change in Momentum
Impulse \( I \) is defined as the change in momentum. The initial momentum \( p_i \) is:
\[ p_i = m(2u) = 2mu \]
The final momentum \( p_f \) is:
\[ p_f = mu \]
Thus, the impulse \( I \) is:
\[ I = p_f - p_i = mu - 2mu = -mu \]

### Step 7: Express Work Done in Terms of Impulse
We can now express the work done in terms of the impulse:
Since \( I = -mu \), we have:
\[ -mu = I \implies mu = -I \]
Substituting this into the work done equation:
\[ W = -\frac{3}{2} mu^2 = -\frac{3}{2} \cdot (-I) \cdot u = \frac{3}{2} I u \]

### Final Answer
Thus, the work done by the force is:
\[ W = \frac{3}{2} I u \]