Consider this:
From A to C, displacement = s
Using SUVAT:
Final Velocity (at C) = u^2 + 2as
Initial Velocity (at C) = u
Average velocity = (u^2 + 2as + u)/2
From A to B, displacement = s/2
Velocity at B = u^2 + 2a x (s/2) = u^2 + as
(u^2 + 2as + u)/2 is NOT always equal to u^2 + as,
Further calculations, assuming they are equal:
(u^2 + 2as + u)/2 = u^2 + as
(u^2)/2 + u/2 + as = u^2 + as
Cancel as out.
(u^2)/2 + u/2 = u^2
This is NOT always true
Method 2:
Acceleration, a, is a given constant. Therefore 2a is a constant. Let 2a be constant m.
Initial velocity, u, is a given constant, therefore u^2 must be constant. Let u^2 be constant c.
Now we are investigating the relationship of velocity at a point, v, and the displacement, s.
Let v^2 = y and let s = x
v^2 = u^2 + 2as
=> y = c + mx
in other words this is
y = mx + c
The relationship between v^2 and s is directly proportional, so if displacement is double v^2 is double.
But if v^2 is double, v is NOT double.