How to derive kinetic energy Ek=1/2mv^2?

Question

Suppose that you don't know the relation,
mechanical work W=F.S.

In this circumastances can you derive the equation Ek=1/2mv^2 (without using the relation W=F.S)?

I think any of the two relation (W=F.S and Ek=1/2mv^2) is axiomatic and they are circular. So that you can't get one of them without using the other. When one of them is defined axiomaticaly you can get the other.
Am I correct? please inform me!!

Answer 1

The definition of work is W=F*s. What do you know about F? Well, you know F=m*a. Insert that into the definition of work:

W=m*a*s

What do you know about the acceleration a? a=d^2s/dt^2. Can you do the rest?