Identify the correct graph represeriting the motion of a particle along a straight line With constant acceleration with zero initial velocity.

Options

A.
B.
C.
D.

Explanation

To solve the problem of identifying the correct graphs representing the motion of a particle along a straight line with constant acceleration and zero initial velocity, we can follow these steps:

### Step 1: Understand the Given Conditions
We are given that the initial velocity (u) is zero and the acceleration (a) is a constant value.

### Step 2: Analyze the Velocity-Time Relationship
Using the equation of motion for velocity:
\[ v = u + at \]
Since \( u = 0 \), the equation simplifies to:
\[ v = at \]
This indicates that the velocity (v) increases linearly with time (t). Therefore, the velocity-time graph will be a straight line starting from the origin (0,0).

### Step 3: Identify the Correct Velocity-Time Graph
From the options provided, we need to identify the graph that represents a straight line passing through the origin. This corresponds to option 1 or option 2.

### Step 4: Analyze the Displacement-Time Relationship
Next, we use the equation for displacement:
\[ s = ut + \frac{1}{2} a t^2 \]
Again, since \( u = 0 \), the equation simplifies to:
\[ s = \frac{1}{2} a t^2 \]
This indicates that the displacement (s) is proportional to the square of time (t²). Therefore, the displacement-time graph will be a parabola that opens upwards and passes through the origin.

### Step 5: Identify the Correct Displacement-Time Graph
From the options provided, we need to find the graph that represents a parabola passing through the origin. This corresponds to option D.

### Conclusion
- The correct velocity-time graph is option 1 (or 2).
- The correct displacement-time graph is option D.

### Final Answer
- Velocity-Time Graph: Option 1 (or 2)
- Displacement-Time Graph: Option D

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