For which of the following graphs the average velocity of a particle moving along a straight line for time interval `(0,t)` must be negative`-`

Options

A.
B.
C.
D.

Explanation

To determine which graph indicates a negative average velocity for a particle moving along a straight line during the time interval \( (0, t) \), we need to analyze the concept of average velocity and how it relates to displacement.

### Step-by-Step Solution:

1. **Understanding Average Velocity:**
Average velocity (\( v_{avg} \)) is defined as the total displacement (\( \Delta x \)) divided by the total time (\( \Delta t \)):
\[ v_{avg} = \frac{\Delta x}{\Delta t} \]
For the interval \( (0, t) \), this can be expressed as:
\[ v_{avg} = \frac{x(t) - x(0)}{t - 0} = \frac{x(t) - x(0)}{t} \]

2. **Condition for Negative Average Velocity:**
For the average velocity to be negative, the displacement must be negative:
\[ x(t) - x(0) < 0 \implies x(t) < x(0) \]
This means that the final position of the particle at time \( t \) must be less than its initial position at time \( 0 \).

3. **Analyzing Graphs:**
We need to look at the position-time graphs provided in the options. The graph that shows the position decreasing over time will indicate a negative displacement.

4. **Identifying the Correct Graph:**
- If the graph shows a downward slope (i.e., the position decreases as time increases), then \( x(t) < x(0) \), leading to a negative average velocity.
- Conversely, if the graph shows an upward slope or remains constant, the average velocity will be zero or positive.

5. **Conclusion:**
After analyzing the graphs, the graph that exhibits a downward slope (indicating that the position of the particle decreases over time) is the correct answer. This corresponds to option A.

### Final Answer:
The graph for which the average velocity of a particle moving along a straight line for the time interval \( (0, t) \) must be negative is **Option A**.