Question

using velocity time graph derive equation of motion UT + 1/2at square

Answer 1

If one considers the fundamental differential definitions of velocity and acceleration (assuming one-dimensional motion) to be v=dx/dt and a=dv/dt, and assume that the acceleration is a constant, then the one-dimensional kinematics equations (as they are called) follow from doing the inverse mathematical operation of integration.

That is, if a=dv/dt and is constant, then a change in velocity is the time integral of the acceleration assuming the problem starts at t=0 , or