Which of the following graph correctly represents velocity-time relationship for a particle released from rest to fall freely under gravity ?

Options

A.
B.
C.
D.

Explanation

To determine the correct velocity-time relationship for a particle released from rest and falling freely under gravity, we can follow these steps:

### Step 1: Understand the initial conditions
- A particle is released from rest, which means its initial velocity (u) is 0 m/s.

### Step 2: Identify the acceleration
- The particle falls freely under the influence of gravity, which means it experiences a constant acceleration (a) equal to the acceleration due to gravity (g). The value of g is approximately 9.81 m/s² (or can be approximated as 10 m/s² for simplicity in calculations).

### Step 3: Use the equation of motion
- The relevant equation of motion that relates velocity (V), initial velocity (u), acceleration (a), and time (t) is given by:
\[ V = u + at \]
- Substituting the known values:
- \( u = 0 \) (initial velocity)
- \( a = g \) (acceleration due to gravity)

This simplifies to:
\[ V = 0 + gt \]
or
\[ V = gt \]

### Step 4: Analyze the relationship
- The equation \( V = gt \) indicates that velocity (V) is directly proportional to time (t). This means that as time increases, the velocity also increases linearly.

### Step 5: Graphical representation
- A linear relationship between V and t, starting from the origin (0,0), indicates that the graph will be a straight line with a positive slope. The slope of this line represents the acceleration due to gravity (g).

### Conclusion
- Therefore, the correct graph that represents the velocity-time relationship for a particle released from rest to fall freely under gravity is a straight line passing through the origin, indicating that velocity increases linearly with time.

### Final Answer
- The correct option is **A**, which shows a straight line graph starting from the origin.

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