First, I'll explain my confusion, which may be enough for someone to clarify my misunderstanding.
Second, to give context, I will give the homework problem (which I've solved successfully) which caused my confusion.
Note: I'll use boldface to denote vector quantities. Also, I am in my third semester of Calculus.
A particle accelerates around a track. Velocity is given by the below equation.
$$ v = v_0 + a_Tt $$
When I substitute in values for this equation to solve homework problems, I use scalar values—in other words, I use magnitudes. I find $|\mathbf{a_T}|$ to use in the above equation.
However, if the equation is written as a sum of vectors:
$$ \mathbf{v} = \mathbf{v_0} + \mathbf{a_T}t $$
Then I take the magnitude of both sides . . .
($\theta$ is the angle between the positive $x$-axis and $\mathbf{a_C}$.)
$$ \begin{align} |\mathbf{v}| &= |\mathbf{v_0} + \mathbf{a_T}t|\\ &\text{. . . algebra . . .}\\ &= \sqrt{|\mathbf{v_0|}^2 + 2|\mathbf{v_0}||\mathbf{a_T}|t + |\mathbf{a_t}|t^2}\\ &\ne |\mathbf{v_0}| + |\mathbf{a_T}|t\ \ \text{*} \end{align} $$
But the starred equation is what I am using for homework problems and I get the answers with no problems.
https://i.stack.imgur.com/mVjCV.png
To summarize my solution:
- Use $\mathbf{a}$ to find $|\mathbf{a_C}|$ and $|\mathbf{a_T}|$.
- Substitute $|\mathbf{a_C}|$ into $a_C = \frac{{v_0}^2}{r}$ and solve for $v_0$.
- Substitute $v_0$ and $\mathbf{a_T}$ into $v_1 = v_0 + a_T t$ and solve for $t_1$ when $v_1 = 0$.
I also run into sign issues because I'm treating the value I get for tangential acceleration as the magnitude of a vector. This means that the value for tangential acceleration is positive, but it needs to subtract from velocity. There is a similar issue with the centripetal acceleration.
The only way to deal with it properly seems to be to treat vectors counter clock-wise around the circle or pointing away from the circle as positive, and other vectors as negative . . . which is akin to treating the circle as a straight line almost . . .
It feels like a hell of a lot of hand-waving, though I am sure that's because I don't understand what is happening.