How did Newton derive equations of motion ?

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How to mathematically derive Newton’s equations of motion ?

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Below is a step by step proof for the same

Acceleration is defined as rate of change of velocity

where u is initial velocity and v is final velocity

       dv          v – u
a = ___  = ________ => v = u + at —————(1)
       dt             t

Velocity is defined as rate of change of displacement

          ds
v = ________ => ds = v.dt ———————–(A)

           dt

From equations (1) and (A) ds = (u+ at)dt

Integrating both sides we get s = ∫(u+at)dt => s = ut + a∫tdt

                    1
=> s = ut + __at^2 ——–(2)
                    2

Substituting the value of t from equation(1)
in equation (2)

        u (v-u)                    (v-u)^2
s = ________ + 1/2*a *   _____
          a                            a^2

         vu – u^2       v^2 + u^2 – 2vu
s = _________ + _________________

            a                                  2a

               2vu – 2u^2 + v^2 + u^2 – 2vu
=> s = ____________________________
                                2a

=> 2as = v^2 – u^2

=> v^2 = u^2 + 2as   —————————-(3)

Equations 1, 2 and 3 prove the 3 equations of motion

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