A force `F_1` acts on a particle so as to accelerate it from rest to a velocity v. The force `F_1` is then replaced by `F_2` which decelerates it to rest

Options

A. `(a)F_(1)`is always equal to `F_(2)`
B. `(b)F_(2)`is greater than `F_(1)`
C. `(c)F_(2)` may be smaller than, greater than or equal to `F_(1)`
D. `(d)F_(2)` cannot be equal to `F_(1)`

Explanation

To solve the problem, we need to analyze the forces acting on a particle as it accelerates and then decelerates.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
- The particle starts from rest, which means its initial velocity \( u = 0 \).
- A force \( F_1 \) is applied to the particle, causing it to accelerate to a velocity \( v \).

2. **Applying Newton's Second Law**:
- According to Newton's second law, the acceleration \( a_1 \) produced by the force \( F_1 \) can be expressed as:
\[ F_1 = m \cdot a_1 \]
- Here, \( m \) is the mass of the particle.

3. **Finding the Final Velocity**:
- The final velocity \( v \) can be related to the acceleration and time using the equation:
\[ v = u + a_1 t \quad \text{(where \( u = 0 \))} \]
- Thus, we have:
\[ v = a_1 t \]

4. **Transition to Deceleration**:
- After reaching the velocity \( v \), the force \( F_1 \) is replaced by a force \( F_2 \) that decelerates the particle to rest.
- This means the final velocity \( v_f = 0 \).

5. **Applying Force \( F_2 \)**:
- The deceleration \( a_2 \) caused by the force \( F_2 \) can be expressed as:
\[ F_2 = m \cdot (-a_2) \]
- The negative sign indicates that the acceleration is in the opposite direction to the motion.

6. **Condition for Coming to Rest**:
- For the particle to come to rest, the force \( F_2 \) must be sufficient to overcome the momentum of the particle. Therefore, \( F_2 \) must be at least equal to or greater than \( F_1 \):
\[ F_2 \geq F_1 \]

7. **Analyzing the Options**:
- **Option A**: \( F_1 \) is always equal to \( F_2 \) - This is incorrect because \( F_2 \) can be greater than \( F_1 \).
- **Option B**: \( F_2 \) is greater than \( F_1 \) - This is a valid condition for deceleration.
- **Option C**: \( F_2 \) may be smaller, greater, or equal to \( F_1 \) - This is incorrect since \( F_2 \) cannot be smaller than \( F_1 \).
- **Option D**: \( F_2 \) cannot be equal to \( F_1 \) - This is incorrect because \( F_2 \) can be equal to \( F_1 \) and still bring the particle to rest.

### Conclusion:
The correct answer is **Option B**: \( F_2 \) is greater than \( F_1 \).