Relationship between distance travelled and time taken
The given problem states that in a straight line motion, the distance travelled is proportional to the square root of the time taken. This means that if the time taken is doubled, the distance travelled increases by a factor of square root of 2.

Acceleration of particle
The problem states that the acceleration of the particle is proportional to some quantity. Let's explore the options given:

1. Velocity: If the acceleration is proportional to velocity, it means that it increases or decreases with the velocity of the particle. This is not a correct option as the problem does not provide any information about the relationship between acceleration and velocity.

2. Velocity squared (v²): If the acceleration is proportional to velocity squared, it means that the acceleration increases or decreases with the square of the velocity of the particle. This is a correct option as it is consistent with the relationship between distance and time provided in the problem.

3. Velocity cubed (v³): If the acceleration is proportional to velocity cubed, it means that the acceleration increases or decreases with the cube of the velocity of the particle. This is not a correct option as it is not consistent with the problem statement.

4. Square root of velocity (√v): If the acceleration is proportional to the square root of velocity, it means that the acceleration increases or decreases with the square root of the velocity of the particle. This is not a correct option as it is not consistent with the problem statement.

Conclusion
Therefore, the correct option is 2)Velocity squared (v²). This means that the acceleration of the particle increases or decreases with the square of its velocity.