The relation `vec(F)=vec(ma)` , cannot be deduced from Newton's second law, if

Options

A. 1. force depends on time
B. 2. momentum depends on time
C. 3. acceleration depends on time
D. 4. mass depends on time

Explanation

To solve the question "The relation \(\vec{F} = m \vec{a}\) cannot be deduced from Newton's second law, if:", we need to analyze the conditions under which this relation holds true.

### Step-by-Step Solution:

1. **Understanding the Equation**: The equation \(\vec{F} = m \vec{a}\) states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. This is a fundamental principle derived from Newton's second law of motion.

2. **Identifying Variables**: In this equation:
- \(\vec{F}\) is the net force acting on the object.
- \(m\) is the mass of the object.
- \(\vec{a}\) is the acceleration of the object.

3. **Considering Mass**: The equation assumes that the mass \(m\) is constant. If the mass of the object changes with time, then the relationship \(\vec{F} = m \vec{a}\) may not hold true because the acceleration could be influenced by the changing mass.

4. **Analyzing the Options**:
- If **force depends on time**: This does not invalidate the equation as forces can change over time while still following the equation.
- If **momentum depends on time**: Momentum can change with time, but this does not negate the relationship since momentum is defined as \(p = mv\) and relates to force.
- If **acceleration depends on time**: Acceleration can change over time, but the equation still holds as it describes the relationship at any instant.
- If **mass depends on time**: This is the critical condition. If mass changes with time, then the equation \(\vec{F} = m \vec{a}\) cannot be deduced from Newton's second law because the standard form assumes constant mass.

5. **Conclusion**: Therefore, the relation \(\vec{F} = m \vec{a}\) cannot be deduced from Newton's second law if mass depends on time.

### Final Answer:
The relation \(\vec{F} = m \vec{a}\) cannot be deduced from Newton's second law if mass depends on time.