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There are 2 things I do not understand in the derivation of kinetic energy from work:

https://www.physicsforums.com/showthread.php?t=111162

(1) W = [itex]\int[/itex][itex]\vec{F}[/itex](t).d[itex]\vec{r}[/itex](t)=

(2) m.[itex]\int[/itex][itex]\frac{d\vec{v}(t)}{dt}[/itex].d[itex]\vec{r}[/itex](t)=

(3) m.[itex]\int[/itex]d[itex]\vec{v}[/itex](t).[itex]\frac{d\vec{r}(t)}{dt}[/itex]=

(4) [itex]\frac{m}{2}[/itex].(v(t1) - v(t0))[itex]^{2}[/itex]

Question I: I don't understand why you can just change the division like that between (2) and (3). I know multiplication and division have the same order but they are calculated from left to right. So why can you do that from (2) to (3)?

http://en.wikipedia.org/wiki/Order_of_operations

Question II: I don't understand why the change in kinetic energy between t1 and t0 is sometimes written as [itex]\frac{1}{2}[/itex].m.v(t1)[itex]^{2}[/itex] - [itex]\frac{1}{2}[/itex].m.v(t0)[itex]^{2}[/itex]

E.g. like in https://www.physicsforums.com/showthread.php?t=194461.

After (4), it should be [itex]\frac{m}{2}[/itex].v(t1)[itex]^{2}[/itex] - m.v(t1).v(t0) + [itex]\frac{m}{2}[/itex].v(t0)[itex]^{2}[/itex] , right?

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# Derivation of kinetic energy

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