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

Options

A.
B.
C.
D.

Answer

To solve the problem of identifying which graph corresponds to an object moving with a constant negative acceleration and a positive velocity, we will follow these steps:

### Step 1: Understand the conditions
We need to analyze the conditions given in the question:
- **Constant negative acceleration**: This means that the acceleration (a) is a negative constant value (e.g., -c).
- **Positive velocity**: This means that the velocity (v) is greater than zero.

### Step 2: Analyze the velocity-time graph
1. **Velocity-time graph**: In a velocity-time graph, if the velocity is decreasing over time while remaining positive, it indicates that the object is experiencing negative acceleration.
- If the graph shows a straight line with a negative slope that remains above the time-axis, it represents constant negative acceleration with positive velocity.

2. **Examine the options**:
- If we have two velocity-time graphs to analyze:
- **Option 2**: If the velocity is increasing, this indicates positive acceleration, which does not meet our condition.
- **Option 4**: If the velocity is decreasing initially but then increases, it suggests a change in acceleration, which also does not meet our condition.

### Step 3: Analyze the velocity-distance graph
1. **Velocity-distance relationship**: The relationship between velocity (v) and distance (x) under constant negative acceleration can be derived from the equation:
\[ a = v \frac{dv}{dx} \]
Given that \( a = -c \) (a constant negative acceleration), we can rearrange and integrate:
\[ v dv = -c dx \]
Integrating both sides gives:
\[ \frac{v^2 - v_0^2}{2} = -cx \]
Rearranging this gives:
\[ v^2 = -2cx + v_0^2 \]
This indicates that the graph of \( v^2 \) versus \( x \) will be a downward-opening parabola with a positive intercept (since \( v_0^2 \) is positive).

2. **Examine the options**:
- **Option 3**: If this graph is a downward-opening parabola with a positive intercept, it satisfies our derived equation and represents the relationship under the conditions of constant negative acceleration and positive velocity.

### Conclusion
Based on the analysis of both the velocity-time graphs and the velocity-distance graph, the correct graph that corresponds to an object moving with a constant negative acceleration and a positive velocity is **Option 3**.

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