Question

Deduce the following equations of motion: (i) s = ut + ( ๐Ÿ/๐Ÿ )at2 (ii) v2 =u 2 + 2as

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Answer

(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 2ย  = ๐‘ข + ๐‘ฃ \over 2

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

Distance, s = Average velocity x Time

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

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

(ii) We know that,

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

Also,ย  ย a = v - u \over t

โŸนย  ย  ย  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

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