A particle moves in a straight line with constant acceleration under a constant force F. Select the correct alternative(s).

Options

A. Power developed by this force varies linearly with time
B. Power developed by this force varies parabolically with time
C. Power developed by this force varies linearly with displacement
D. Power developed by this force varies parabolically with displacement

Answer

To solve the problem, we need to analyze the motion of a particle moving in a straight line with constant acceleration under a constant force \( F \). We will derive the relationships between power, velocity, and displacement.

### Step-by-Step Solution:

1. **Understanding the Basics**:
- A particle is moving with constant acceleration \( a \) under a constant force \( F \).
- According to Newton's second law, \( F = ma \), where \( m \) is the mass of the particle.

2. **Power Definition**:
- Power \( P \) is defined as the rate at which work is done or energy is transferred. Mathematically, it can be expressed as:
\[ P = F \cdot v \]
where \( v \) is the velocity of the particle.

3. **Velocity Expression**:
- From the equations of motion, the velocity \( v \) of the particle at time \( t \) can be expressed as:
\[ v = u + at \]
where \( u \) is the initial velocity.

4. **Substituting Velocity into Power**:
- Substituting the expression for \( v \) into the power equation gives:
\[ P = F \cdot (u + at) \]
- This shows that power is directly proportional to time \( t \) since \( F \) and \( u \) are constants.

5. **Displacement Expression**:
- Another equation of motion relates displacement \( s \) to time:
\[ s = ut + \frac{1}{2}at^2 \]

6. **Finding Power in Terms of Displacement**:
- From the equation \( v^2 = u^2 + 2as \), we can express \( v \) in terms of displacement:
\[ v = \sqrt{u^2 + 2as} \]
- Substituting this into the power equation gives:
\[ P = F \cdot \sqrt{u^2 + 2as} \]

7. **Analyzing Power with Displacement**:
- As \( s \) increases, \( P \) will vary as the square root of \( s \). Therefore, we can say:
\[ P^2 \propto s \]
- This indicates that power varies parabolically with displacement \( s \).

### Conclusion:
- The correct alternatives are:
- Power developed by this force varies linearly with time.
- Power developed by this force varies parabolically with displacement.