Can someone please tell me what I have done wrong here? I can't see what is wrong and have asked many people far smarter than me who also can't spot the error.
closed as off-topic by David Z♦ Sep 15 at 10:42
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You're using $v$ in two different ways. In the first few equations, $v$ means the final velocity after a period of uniform acceleration, assuming one starts from rest. In the equation $v = s/t$, $v$ means the average velocity instead. That's half as much, which is why you're off by $2$.
This is a general warning for learning physics. In other high school courses, if you just take the formulas you're given and mash them together, without worrying what the variables mean, you can get the right answer. In physics, you really have to understand what you're doing. Many formulas will have the letter $v$ in them, but the meaning of that letter will be different, because there can be many different important speeds in most problems.
In the last step $v^2=2v^2$ means $ v=0$. So you cannot divide both sides by $v^2$ because that is division by 0. That is why you get nonsense.