If the displacement of an object is proportional to the square of the time taken then the body is moving with uniformly accelerated motion as it will follow Newton's second equation of motion for a particular initial velocity, which can be given by, s = ut + 1/2 at²s = ut + 21at²

Explanation:

When the displacement of an object is proportional to the square of time, it means that the object is undergoing uniform acceleration. This can be understood by examining the relationship between displacement, velocity, and time.

- Displacement: Displacement refers to the change in position of an object. It is a vector quantity and has both magnitude and direction.

- Velocity: Velocity is the rate of change of displacement with respect to time. It is also a vector quantity and has both magnitude and direction.

- Acceleration: Acceleration is the rate of change of velocity with respect to time. It is also a vector quantity and has both magnitude and direction.

Proportional to Square of Time:

When the displacement of an object is proportional to the square of time, it can be expressed as:

s = kt^2

where s represents displacement, t represents time, and k is a constant.

Analysis:

1. Displacement vs. Time: If we plot a graph of displacement versus time, we will obtain a parabolic curve. This is because the displacement is proportional to the square of time.

2. Velocity vs. Time: To find the velocity, we need to differentiate the displacement equation with respect to time.

v = d/dt (kt^2) = 2kt

The velocity is directly proportional to time, indicating that the object is undergoing uniform acceleration.

3. Acceleration vs. Time: To find the acceleration, we need to differentiate the velocity equation with respect to time.

a = d/dt (2kt) = 2k

The acceleration is constant and does not depend on time. Hence, the object moves with uniform acceleration.

Conclusion:

Therefore, when the displacement of an object is proportional to the square of time, the object moves with uniform acceleration (option B). This is because the velocity is directly proportional to time, and the acceleration is constant.