I have read that there is a relation between the theorem of Pythagoras and the fact that kinetic energy is proportional to the square of velocity.
But I can't find on internet any proof or further information about this.
Any answer, comment or link would be welcome
I can't recall where I read this, it is just a joke.
Suppose that a particle with mass $m=2$ is initially at rest. Accelerate the particle along the $x$-axis until it reaches a velocity $v_x$. The work done is $W_1 =v_x ^2$. Now accelerate the particle along the $y$ axis until it reaches velocity $v_y$ along the $y$ direction. The work done doesn't depend on $v_x$, since you are working at right angles to $v_x$. So the work done is just $W_2 =v_y ^2$. Now, the total work done is $W=W_1+W_2$. This work must equal $W=v^2$, the square of the final velocity.
Conclusion: $$v^2 = v_x ^2+v_y^2.$$
We can prove mostly geometrically that kinetic energy must be a function of $\vec{v}^2 \equiv \vec{v}\cdot \vec{v} = v_x^2+v_y^2+v_z^2$, but we need extra assumptions to prove that it is directly proportional to $\vec{v}^2$.
Define the kinetic energy to be a scalar quantity that depends on velocity, is conserved in elastic collisions, and is invariant under rotations of reference frame. To find the form of this quantity we need to find scalar invariant functions formed from $\vec{v}$ alone. We can reason geometrically that any such number is a function of $\vec{v}^2$. This is because rotations preserve lengths and angles, but with only one input vector the only invariant quantity is the length of the vector, given by $\sqrt{\vec{v}^2}$.
However, geometry alone is not enough to give us $E\propto \vec{v}^2$. This is clear because in classical mechanics $E\propto \vec{v}^2$, but in special relativity $E \propto \frac{1}{\sqrt{1-\vec{v}^2/c^2}}$. Both are functions of $\vec{v}^2$ because both theories are invariant under rotations, but they have different functional forms.
(The different forms are so that energy can be conserved in elastic collisions in all reference frames that differ by a boost in velocity. The two theories handle boosts differently, so the energies have different forms.)
