Question

Power and work
\[ W= \int P dt = \int Fv dt= m \int \frac{dv}{dt} vdt=m \int vdv= \frac{1}{2}m(v_1^2-v_0^2) \]

Is this true?:
-Did I oversimplify somewhere?
-Is this unphysical?

Answer 1

\[ W= \int\limits P dt = \int\limits Fv dt= m \int\limits \frac{dv}{dt} vdt=m \int\limits vdv= \frac{1}{2}m(v_1^2-v_0^2) \]

Answer 2

it's fine.

Answer 3

correct

Answer 4

Similarly, \[W=\int\limits_{}^{}Fdx=\int\limits_{}^{}madx=m \int\limits_{}^{} \frac{ dv }{ dt } dx=m \int\limits_{}^{} \frac{ dv }{ dx }\frac{ dx }{ dt } dx = m \int\limits_{}^{} v dv=\frac{ 1 }{ 2 }mv^2\]

Answer 5

@Kainui That's what I did.

Answer 6

No, you did something slightly different. You used power, I did not.