Question

proof that, 2as =v^2 - u^2

Answer 1

This equation can be proved as follows:
v = u + at therefore t = (v-u)/a but s = ut + ½ at2 and so

s = ut + ½ a([v-u]/a)2 therefore: 2s = 2u(v-u)/a + (v2 – 2uv + u2)/a

So: 2as = 2uv – 2u2 + v2 – 2uv + u2 and so v2 = u2 + 2as

Answer 2

change in kinetic energy = 1/2 m v² - 1/2 m u²
Force on the body =  m a
distance travelled = displacement = s

So   energy conservation (work energy theorem)
          1/2  m (v² - u²) = m a s
           so  2as  = v² - u²
============================

v² - u² = (v - u) ( v + u)
   = a t (u + u + at)
   = 2 a (u + 1/2 a t²)
    = 2 a s