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NeedBetterName
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Homework Statement
Derive the formula for final velocity, [itex]v_2[/itex], without time included in said formula.
[itex]\overrightarrow{a}[/itex]: Known
[itex]\overrightarrow{s}[/itex]: Known
[itex]v_1[/itex]: Known
[itex]\Delta t[/itex]: Unknown (can be calculated, but defeats the purpose)
[itex]\overrightarrow{v_2}[/itex]: Unknown and to be calculated
Homework Equations
[tex]
\begin{align*}
\Delta t &= \frac{\overrightarrow{s}}{\overrightarrow{\Delta v}} \\ \\
\overrightarrow{a} &= \frac{\overrightarrow{\Delta v}}{\Delta t}
\end{align*}
[/tex]
The Attempt at a Solution
Since I've always disliked memorizing formulas, I just figure out how they're derived and then if I forget the formula, I can get it regardless; so I went about trying to get the formula as follows:
[tex]
\begin{align*}
\overrightarrow{a} &= \frac{\overrightarrow{\Delta v}}{\Delta t} \\
&= \frac{\overrightarrow{\Delta v}}{\frac{\Delta \overrightarrow{s}}{\Delta \overrightarrow{v}}} \\
&= \frac{\Delta \overrightarrow{v}^2}{\Delta \overrightarrow{s}} \\
&= \frac{(v_2 - v_1)^2}{\Delta \overrightarrow{s}} \\
\overrightarrow{a} \Delta \overrightarrow{s} &= (v_2 - v_1)^2 \\
&= v_2^2 - 2v_2v_1 - v_1^2 \\ \\
v_2^2 &= -2v_2v_1 - v_1^2 + \overrightarrow{a} \Delta \overrightarrow{s}
\end{align*}
[/tex]
I ended up with a result that different than the formula taught in my class. I can't remember the formula taught in class, as I wasn't actually there for the lesson, but I vaguely remember what the formula was like, when I saw it later. I foolishly forgot to write it down, so now I'm not too sure, but I think it was similar to [itex]\v_2^2 = v_1^2 + \overrightarrow{a}\Delta \overrightarrow{s}[\itex].
4. What I really want to know
I basically need to know 4(ish) things: Is my derived formula okay? If not, where'd I go wrong? What's the formula similar to what I listed above (or is it correct; this question also isn't too big of a deal since I can get it tomorrow)? And if the second formula is correct (or the correct, similar one has been provided), how is it derived?
EDIT: I remember the formula in class, which is [itex]v_2^2 = v_1^2 + 2 \overrightarrow{a} \Delta \overrightarrow{s}[/itex]. How is this derived?
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