I have seen the derivation of the kinetic energy equation using
And I can see how this works, however if you try to do this without thinking about velocity, and only thinking about the rate of change of distance, and the rate of change of rate of change of distance, then the derivation doesn't work, as shown below.
F = Force
M = Mass
x = Distance
Force = mass * acceleration and
Energy = the integral of force with respect to distance:
F = M * x''
E = integral ( F .dx )
sub F into E = integral ( F .dx )
E = integral ( M * x'' .dx )
E = M * integral ( x'' .dx )
E = M * x'
Which isn't true. E should equal 0.5*M*(x')^2
Why does this not work?