# How did Newton derive equations of motion ?

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