My example; for mathematical simplicity lets look at one
dimensional motion; the primed frame (K') is the non-inertial
frame. The origins coincide at t=0 and the acceleration of K'
relative to K is a*i_hat, where a is a constant. Furthermore, K'
has zero speed relative to K at t=0.
Initial conditions:
x(0) = d
x_dot(0) = 0
no real forces present in the inertial frame (K)
Problem find the final speed v'_f of the object as measured by K'
when x'(t_f) = 0.
Method I (standard Newtonian method)
work-energy them applied in inertial frame:
no forces implies v_f = 0
transform v_f into the K' system
v'_f = v_f minus speed of K' relative to K at t=t_f . . .(1)
t_f is solution to
1/2 a t_f*t_f = d
from standard constant acceleration equation
applied to motion of the origin of K'
initial conditions:
x'(0) = d
v'_i(0) = x'_dot(0)= 0
and F'= ma' = -ma, the apparent force term
work-energy theorem, in non-inertial frame:
0.5*m*v'_f(squared) = Int(F'*dx')=Int(-ma*dx')
divide by m on both sides and integrate to get
0.5*v'_f(squared) = -a*(0 - d)
Hence v'_f(squared) = 2*a*d
In full agreement with (2) above, an example of
my statement that Marlow and I will get the answers.
This kind of calculation is all I've ever claimed.