To describe the motion of a particle acted upon by the given force equations, we can analyze the forces and their graphical representations step by step.
### Step 1: Identify the Force Equations
The force equations given are:
1. \( F = 3x + 3 \)
2. \( F = -3x - 3 \)
3. \( F = -3x + 3 \)
4. \( F = -3x - 3 \)
### Step 2: Analyze Each Force Equation
For each force equation, we will analyze how the force varies with position \( x \).
1. **For \( F = 3x + 3 \)**:
- When \( x = 0 \), \( F = 3 \).
- When \( x = 1 \), \( F = 6 \).
- The force increases linearly as \( x \) increases.
- **Graph**: A straight line with a positive slope, starting from \( (0, 3) \).
2. **For \( F = -3x - 3 \)**:
- When \( x = 0 \), \( F = -3 \).
- When \( x = 1 \), \( F = -6 \).
- The force decreases linearly as \( x \) increases.
- **Graph**: A straight line with a negative slope, starting from \( (0, -3) \).
3. **For \( F = -3x + 3 \)**:
- When \( x = 0 \), \( F = 3 \).
- When \( x = 1 \), \( F = 0 \).
- The force decreases linearly and crosses the x-axis.
- **Graph**: A straight line with a negative slope, starting from \( (0, 3) \) and crossing the x-axis at \( (1, 0) \).
4. **For \( F = -3x - 3 \)** (same as the second):
- The analysis is the same as the second equation.
- **Graph**: A straight line with a negative slope, starting from \( (0, -3) \).
### Step 3: Determine the Motion of the Particle
The motion of the particle can be described based on the force acting on it:
- **Positive Force**: When the force is positive (like in the first equation), the particle will accelerate in the positive direction.
- **Negative Force**: When the force is negative (like in the second and fourth equations), the particle will accelerate in the negative direction.
- **Equilibrium**: In the third equation, the force becomes zero at \( x = 1 \), indicating a point of equilibrium where the particle may oscillate.
### Conclusion
The particle's motion will vary depending on the force acting on it as described by the equations. The graphical representation helps visualize how the force changes with position, indicating the direction and nature of the particle's motion.
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