Deriving Big 5 Motion Laws Non-Calculus and Calculus

[quincyboy7]

"Simple" derivations

I set out trying to derive the big 5 motion laws not only regularly, but calculus-style:

dv=adt
Integration yields v=at + C, and thus we get v=at+v0.
Non-calculus method is just using the definition of a=deltav/deltat and is trivial.

dx=vdt
x=(v0+at)dt
x=v0t + 1/2at^2 + C, and thus x=v0t + 1/2 at^2 + x0.

dx=v0dt
x=(v-at)dt
x=vt-1/2at^2+C, and thus x=vt-1/2at^2 + x0.

v_avg=xt, and v_avg=(v0+v)/2...i get that because of simple math and graphing with constant acceleration, but is there a calculus way to prove this?

What are the non-calculus derivations for the two i just mentione above.
Also, the last one has completely stumped me...(deltav^2=2ax), both non-calculus and calculus methods. If you have any hints/derivations with which to enlighten me, that would be much appreciated. Thanks.

 

[quincyboy7]

I set out trying to derive the big 5 motion laws not only regularly, but calculus-style:

dv=adt
Integration yields v=at + C, and thus we get v=at+v0.
Non-calculus method is just using the definition of a=deltav/deltat and is trivial.

dx=vdt
x=(v0+at)dt
x=v0t + 1/2at^2 + C, and thus x=v0t + 1/2 at^2 + x0.

dx=v0dt
x=(v-at)dt
x=vt-1/2at^2+C, and thus x=vt-1/2at^2 + x0.

v_avg=xt, and v_avg=(v0+v)/2...i get that because of simple math and graphing with constant acceleration, but is there a calculus way to prove this?

What are the non-calculus derivations for the two i just mentioned above.
Also, the last one has completely stumped me...(deltav^2=2ax), both non-calculus and calculus methods. If you have any hints/derivations with which to enlighten me, that would be much appreciated. Thanks.

Somewhat solved. a=dv/dt=(dv/dx)(dx/dt)=vdv/dx. Then adx=vdv and ax=v^2/2 + C and so the show goes on. Wicked relieved, but still would appreciate any calculus answers to why vavg=1/2 (v0+v) and a non-calculus answer to the last rule still.