Deduce the following equations of motion: (i) s = ut + ( 𝟏/𝟐 )at2 (ii) v2 =u 2 + 2as

(i) Consider a body which starts with initial velocity u and due to uniform acceleration a, its final velocity becomes v after time t. Then, its average velocity is given by

Average velocity = Initial\ velocity + Final\ velocity \over 2  = 𝑢 + 𝑣 \over 2

∴ The distance covered by the body in time t is given by

Distance, s = Average velocity x Time

or  {u + v \over 2} \times t or s = {u + (u + at) \over 2} \times t

∴   s = 2ut + at^2 \over 2     or  s = ut + {1 \over 2} at^2

(ii) We know that,

s = ut + {1 \over 2} at^2

Also,   a = v - u \over t

⟹      t = {v - u \over a}

Putting the value of t in (1), we have

s = u({v - u \over a}) + {1 \over2}a({v - u \over a})^2

s = {uv - u^2 \over a} + {v^2 + u^2 - 2uv \over 2a}

2as = 2uv – 2u² + v² + u² – 2uv

v² – u² = 2as