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billard

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- TL;DR Summary
- Issue deriving 1/2 *mv^2 from some pre-assumed equations

Hello! I am new to the differential version of classical physics, and I am trying to work how to derive kinetic energy from some pre-assumed equations:

Assume that we know: ##\ddot{z} = 0## and ##m\ddot{\textbf{r}} \cdot \dot{\textbf{r}} = 0##This results in $$\frac{1}{2}m\dot{r}^2 = W = const.$$

How is the kinetic energy given here with our pre-assumptions? I am sure this is very simple, forgive me, I am a beginner.

Assume that we know: ##\ddot{z} = 0## and ##m\ddot{\textbf{r}} \cdot \dot{\textbf{r}} = 0##This results in $$\frac{1}{2}m\dot{r}^2 = W = const.$$

How is the kinetic energy given here with our pre-assumptions? I am sure this is very simple, forgive me, I am a beginner.