Which graph corresponds to an object moving with a constant negative acceleration and a positive velocity ?

Options

A.
B.
C.
D.

Answer

To determine which graph corresponds to an object moving with a constant negative acceleration and a positive velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Terms**:
- **Constant Negative Acceleration**: This means that the object is slowing down. The acceleration is negative, which indicates that it acts in the opposite direction to the velocity.
- **Positive Velocity**: This means that the object is still moving in the positive direction, but its speed is decreasing due to the negative acceleration.

2. **Mathematical Representation**:
- Let the acceleration be \( a = -c \) (where \( c \) is a positive constant).
- According to the kinematic equations, we can relate acceleration, velocity, and displacement.

3. **Using the Relationship Between Velocity and Displacement**:
- We can use the equation \( v \frac{dv}{dx} = a \).
- Substituting \( a = -c \), we have:
\[ v \frac{dv}{dx} = -c \]

4. **Separating Variables**:
- Rearranging gives:
\[ v \, dv = -c \, dx \]

5. **Integrating Both Sides**:
- Integrate both sides:
\[ \int v \, dv = \int -c \, dx \]
- This results in:
\[ \frac{v^2}{2} = -cx + k \]
- Rearranging gives:
\[ x = -\frac{v^2}{2c} + \frac{k}{c} \]

6. **Analyzing the Equation**:
- The equation \( x = -\frac{v^2}{2c} + \frac{k}{c} \) is a quadratic equation in terms of \( v \) and represents a downward-opening parabola.
- As \( v \) decreases (due to negative acceleration), \( x \) will also change, indicating that the object is moving in the positive direction but slowing down.

7. **Graphical Interpretation**:
- The graph will show a downward-opening parabola, indicating that as the displacement \( x \) increases, the velocity \( v \) decreases.
- Therefore, the correct graph corresponds to a curve that starts from a positive value and slopes downwards.

8. **Conclusion**:
- The graph that matches this description is option 3, which shows a downward trend in velocity as displacement increases.