Explanation:

Displacement and Time Relationship:
When it is stated that displacement is proportional to the square of time, it means that the displacement (d) is directly proportional to the square of the time (t^2). Mathematically, this relationship can be expressed as:

d ∝ t^2

This indicates that as the time increases, the displacement will increase at a rate proportional to the square of that time.

Acceleration and Displacement Relationship:
Acceleration (a) is the rate at which the velocity of an object changes with time. It is defined as the change in velocity divided by the change in time. Mathematically, acceleration can be expressed as:

a = (v - u) / t

Where 'v' is the final velocity, 'u' is the initial velocity, and 't' is the time taken.

Uniform Acceleration:
Uniform acceleration refers to a situation where the acceleration of an object remains constant over a given time period. This means that the object's velocity changes by the same amount in equal intervals of time.

Relationship between Displacement and Uniform Acceleration:
If the displacement is directly proportional to the square of time (d ∝ t^2), then its derivative, velocity, would be directly proportional to time (v ∝ t). Taking the derivative of velocity, which is acceleration, we get:

a = dv/dt

Since v ∝ t, the derivative of v with respect to t is a constant value. Therefore, the acceleration remains constant (uniform) when the displacement is proportional to the square of time.

Example:
Let's consider a scenario where an object starts from rest (u = 0) and undergoes uniform acceleration. If the displacement of the object is given by d = kt^2, where k is a constant, we can find the acceleration.

Taking the derivative of displacement with respect to time, we get:

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

Now, taking the derivative of velocity with respect to time, we get:

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

As we can see, the acceleration is a constant value of 2k, indicating uniform acceleration.

In conclusion, when displacement is proportional to the square of time, the acceleration is uniform because the derivative of velocity with respect to time is constant. This relationship holds true in scenarios where the object's displacement follows a quadratic function of time.