A little deeper

Newton and Energy.

The relation between force and work and kinetic energy comes for free with some combination of one of our kinemetics equations and Newton’s second law.

From Equation 6.5, we remember:

v^2 = v_0^2 + 2ax

So there’s the acceleration, just sitting there. What do do with it? Isolate it:

a=\frac{v^2 - v_0^2}{2x}

Why go to the defining relationship from Newton and plug it in:

\begin{align*} F &=ma \\ & = m \left(\frac{v^2 - v_0^2}{2x}\right) \\ Fx &= \frac{m}{2}\left( v^2 - v_0^2 \right) \\ Fx &= \dfrac{1}{2}mv^2 - \dfrac{1}{2}mv_0^2 \\ Fx &= \Delta K = W \end{align*}

Here’s the