Velocity at the Midpoint of a uniform acceleration

In this post, we will derive the Average Velocity Formula at the midpoint of a straight line distance traversed by a moving body with a uniform acceleration.

In this specific case, say, a body starts with an initial velocity u from point A and completes the path at point B with a velocity v. Distance between A and B = length of the path traversed = S.
For this case, we need to find out the average velocity formula at the midpoint M of AB distance.
When the body moves to the midpoint then the distance traveled = AM = s/2
Say, the velocity at M is V_mid.

v² = u² + 2 a s
=> s = (v² – u²)/ (2 a) ………………. (1)

V_mid² = u² + 2 a (s/2)
=> V_mid² = u² + a s
=> V_mid² = u² + a(v² – u²)/ (2 a)
=> V_mid² = u² + (v² – u²)/(2)
=>V_mid² = (u² + v²)/ (2)
=>V_mid = [(u² + v²)/2]^1/2

So, the velocity at the midpoint can be expressed by the formula: V_mid = [(u² + v²)/2]^1/2