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