The equation "x = ut + (1/2)at^2" is derived using average velocity because it is meant to describe the motion of an object with constant acceleration over a specific time interval. This equation assumes that the acceleration is constant throughout that interval.
When using average velocity, we assume that the acceleration is constant during the entire time interval but that the velocity of the object changes uniformly from its initial velocity to its final velocity. This assumption allows us to simplify the calculations and arrive at a useful equation for position.
Using the actual, instantaneous velocity at some specific time to derive the equation "x = ut + at^2" would not accurately represent the motion of an object with constant acceleration over a given time interval. The equation "x = ut + at^2" is a simplified form of the more general equation for position when acceleration is constant, which is "x = x0 + v0t + (1/2)at^2", where x0 is the initial position and v0 is the initial velocity.
By using average velocity instead of instantaneous velocity, we assume that the velocity is changing uniformly over the time interval, simplifying the calculation and providing a reasonable approximation for the object's position. This equation is particularly useful when the acceleration is constant and the motion can be described in terms of average values over a specific time interval.
If you need to find the exact position at a specific time, it is important to use the more general equation for position that incorporates the initial position and velocity.