I wanted to clarify if kinetic energy is always positive. Since $KE = 1/2 mv^2$, and $m$ and the square of $v$ is positive. I assume as such.
Given that I have a scenario where an object which was travelling at a positive velocity in a certain direction (we take this direction as positive), reverses the direction and travels at a negative velocity similar in magnitude. The $KE$ final and initial would be equal due to $KE$ always being positive. Now, the change in $KE$ would be $0$, since both values are equal positive values. However, that is not the case as $W = Fd\cos(\theta)$. And clearly, there must be a force enacted to change the velocity of the object. Assume there is a distance present over which the force acts. Now, the work should be negative as $\cos(\theta)$ is negative.
Given this, I don't understand what I am supposed to do when writing $KE$ or $W$. If $KE$ is negative relative to something else, would I write $KE = - 1/2 mv^2$?
$W=Fd\cos(\theta)$ or $W=Fs$. In the $\cos\theta$ version, are $F$ and $d$ magnitude/s, vector/s or scalar/s? In the $W=Fs$ version, I have the same question.
I read somewhere that scalars are simply values. In that case, are scalars magnitudes with signs, rather than just magnitudes? Or is a magnitude being an absolute value a misconception?
Say that I have a velocity time graph, and the velocity changes at a certain point. These points of change can not be differentiated can they? Would the end point's slope be undefined as well?