* Displacement and Acceleration: Displacement (change in position) is directly related to acceleration. If an object is accelerating, its displacement increases over time.

* Proportional to the Square of Time: When displacement is proportional to the square of time, it means the displacement increases at an increasing rate. This is a hallmark of constant acceleration.

Mathematical Explanation:

Let's say the displacement (s) is proportional to the square of time (t):

* s ∝ t²

This can be written as:

* s = kt² (where k is a constant of proportionality)

Now, let's look at the relationship between displacement, velocity (v), and acceleration (a):

* Velocity (v) is the rate of change of displacement: v = ds/dt

* Acceleration (a) is the rate of change of velocity: a = dv/dt

Differentiating the equation s = kt² twice with respect to time, we get:

1. Velocity: v = ds/dt = 2kt

2. Acceleration: a = dv/dt = 2k

Since the acceleration (a) is a constant (2k), the body is moving with uniform acceleration.

Example:

Imagine a ball falling freely under gravity. Its displacement is proportional to the square of the time it falls. This is because gravity provides a constant acceleration to the ball.