derivation of s=ut+1/2at2?

Question

s=average velocity *time
average velocity =x2-x1/t2-t1
but to derive the above formula why do we use average velocity as u+v/2 and not delta x/delta t?

Answer 1

Integrate d^2 s/dt^2=a twice. Assume a is constant. Keep your integration constants.

Answer 2

Because velocity is changing. Think about it, at the beginning of a trip where you are speeding up from rest, your instantaneous velocity is zero. At the end of the acceleration it is much faster. If you simply used delta x/delta t you would be skewed towards the velocities that were slower because you spend more time traveling the shorter amounts of distance in the first part of your journey and less time traveling longer amounts of distance in the final part of your journey. The way to resolve this changing velocity problem is to invent calculus (that's what Newton did). The way we "fudge" it is to tell you to use u + v/2 instead of delta x/delta t.

In point of fact, this only works when your acceleration is constant. If your acceleration is changing through your trip, this equation fails as well.