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The formula for Kinetic Energy $T$ is $T=\frac{1}{2}mv^2$

I can derive this using Calculus so I don't need to be shown that.

I can also derive it using Algebra but have some issues as follows.

$$T=Fs=mas$$

From the equation $v^2=u^2+2as$ we get $as=\frac{v^2}{2}$

Subbing this into $T=mas=m(\frac{v^2}{2})=\frac{1}{2}mv^2$

Great, but was this algebraic derivation legitimate?

We also have the equation $v=u+at$ which gives us $a=\frac{v}{t}$

Subbing that in gives us $T=mas=m(\frac{v}{t})s=mv^2$

Why does the first substitution work yet the second does not? Both substitutions come from equations which assume constant acceleration.

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closed as off-topic by Feynmans Out for Grumpy Cat, stafusa, Bill N, sammy gerbil, ZeroTheHero Oct 12 '17 at 20:50

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    $\begingroup$ The formulas you use are valid only for constant acceleration (IOW, you do need calculus to derive things in general). But $v = s/t$ is only valid for zero acceleration. $\endgroup$ – NickD Oct 9 '17 at 17:28
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    $\begingroup$ Voting to close as too personal "check my work" type questions do not come under the domain of PSE as far as I understand. Nick's comment anyway resolves OP's question and as I said, this seems like a too personal "check my work" question - thus, it mostly won't be of a broader audience's scientific interest. $\endgroup$ – Feynmans Out for Grumpy Cat Oct 9 '17 at 17:58
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$s=vt$ is true only for constant speed. Note also that the other equations you are using are only true for constant acceleration.

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