A particle is thrown above, then correct `v-t` graph will be

Options

A.
B.
C.
D.

Explanation

To solve the problem of determining the correct velocity-time (v-t) graph for a particle thrown upwards, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion**:
- When a particle is thrown upwards, it initially moves against the force of gravity. As it ascends, its velocity decreases until it reaches a maximum height where the velocity becomes zero.

2. **Initial Conditions**:
- Let’s denote the initial velocity of the particle as \( u \) (which is positive since it is thrown upwards).
- At \( t = 0 \), the velocity \( v = u \).

3. **Velocity Decrease**:
- As the particle rises, the acceleration due to gravity (\( g \)) acts downwards, causing the velocity to decrease.
- The relationship between initial velocity, final velocity, acceleration, and time can be described by the equation:
\[ v = u - gt \]
- This equation indicates that the velocity decreases linearly with time until it reaches zero.

4. **Maximum Height**:
- At the maximum height, the velocity \( v \) becomes zero. This occurs at a certain time \( t = t_{max} \), where:
\[ 0 = u - gt_{max} \implies t_{max} = \frac{u}{g} \]

5. **Velocity After Maximum Height**:
- After reaching the maximum height, the particle starts to fall back down. The velocity will now be negative (since it is moving in the opposite direction).
- The velocity as it falls can be described by:
\[ v = 0 + gt \quad (\text{for } t > t_{max}) \]
- This indicates that the velocity increases in magnitude but remains negative as the particle falls back down.

6. **Graph Representation**:
- The v-t graph will start at \( u \) (positive), decrease linearly to zero at \( t_{max} \), and then continue to decrease (becoming negative) as the particle falls back down, increasing in magnitude.

7. **Conclusion**:
- The correct v-t graph will show a linear decrease from \( u \) to 0, followed by a linear increase in the negative direction (indicating increasing speed downwards).

### Final Answer:
The correct v-t graph for a particle thrown upwards is represented by option A, which shows the described behavior.