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. `F_1` must be equal to `F_2`
B. `F_1` may be equal to `F_2`
C. `F_1` must be unequal to `F_2`
D. none of these

Answer

To solve the problem, we will analyze the motion of the particle under the influence of the forces \( F_1 \) and \( F_2 \) using Newton's laws of motion.

### Step-by-Step Solution:

1. **Identify the Initial Conditions:**
- The particle starts from rest and is accelerated to a velocity \( v \) by the force \( F_1 \).
- The mass of the particle is denoted as \( M \).

2. **Acceleration During the First Phase (A to B):**
- Since the particle starts from rest, the initial velocity \( u = 0 \) and the final velocity \( v \) is reached.
- Let \( t_1 \) be the time taken to reach velocity \( v \).
- Using the formula for acceleration:
\[ a = \frac{v - u}{t_1} = \frac{v - 0}{t_1} = \frac{v}{t_1} \]

3. **Applying Newton's Second Law for \( F_1 \):**
- According to Newton's second law, the force \( F_1 \) can be expressed as:
\[ F_1 = M \cdot a = M \cdot \frac{v}{t_1} \]

4. **Acceleration During the Second Phase (B to C):**
- In this phase, the particle is decelerated to rest, meaning the final velocity is \( 0 \) and the initial velocity is \( v \).
- Let \( t_2 \) be the time taken to come to rest.
- The acceleration during this phase can be calculated as:
\[ a' = \frac{0 - v}{t_2} = \frac{-v}{t_2} \]

5. **Applying Newton's Second Law for \( F_2 \):**
- The force \( F_2 \) acting on the particle can be expressed as:
\[ F_2 = M \cdot a' = M \cdot \left(\frac{-v}{t_2}\right) = -M \cdot \frac{v}{t_2} \]

6. **Comparison of Forces \( F_1 \) and \( F_2 \):**
- We have:
\[ F_1 = M \cdot \frac{v}{t_1} \]
\[ F_2 = -M \cdot \frac{v}{t_2} \]
- Since \( F_1 \) is positive (acting in the direction of motion) and \( F_2 \) is negative (acting against the direction of motion), we can conclude that:
\[ |F_1| = M \cdot \frac{v}{t_1} \quad \text{and} \quad |F_2| = M \cdot \frac{v}{t_2} \]

7. **Conclusion:**
- Without knowing the relationship between \( t_1 \) and \( t_2 \), we cannot definitively say whether \( F_1 \) is equal to \( F_2 \) or not. Thus, the correct interpretation is that \( F_1 \) may or may not be equal to \( F_2 \).

### Final Answer:
Therefore, the conclusion is that \( F_1 \) may be equal to \( F_2 \), but we cannot assert that they must be equal.

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