Galileo did pioneering investigations on the laws of motion and derived the proportionalities shown below.

With the beneft of calculus, Newton, who was born in 1642, the year that Galileo died, developed these proportionalities into the familar equations of motion.

Galileo

Newton

Condition

Proportion

Proportionality

Constant

Equation

Distance travelled s at time t assuming continuous, uniform motion (constant velocity v )

s ∝ vt

1

s = vt

Distance travelled s at time t after uniform acceleration a from rest

s ∝ vt

½

s = ½ vt

Velocity v at time t after uniform acceleration a from rest

v ∝ t

a

v = at

Velocity v at time t after uniform acceleration a from initial velocity u

v = u + at

Distance travelled s at time t after uniform acceleration from rest

s ∝ t^{2}

½ a

s = ½ a t^{2}

Distance travelled s at time t after uniform acceleration a from initial velocity u