Yes. The kinetic energy of a bullet is zero in the frame of an observer who is at rest relative to the bullet, non-zero for an observer who is at rest relative to the target of the bullet.
Ok, Right. The statement "neglecting a constant" is my mistake. I clarify my purpose of the question: Newton's laws are only valid in inertial framework. I like to know whether energy formalism is valid in non-inertial framework or not? In other words, can one solve the problems exactly, using conservation of energy in non-inertial framework?
[itex]\int_{t_0}^{t_1}\vec{F}(t)\cdot\vec{v}(t)dt = \frac{1}{2}m v^2(t_1) - \frac{1}{2}m v^2(t_0)[/itex] is valid in frames where [itex]\vec{F}(t) = m \frac{d\vec{v}(t)}{dt}[/itex] That is, in inertial frames. You still can use it in non-inertial frames IF you add "inertial forces". [itex]\int_{t_0}^{t_1}\vec{F}(t)\cdot\vec{v}(t)dt = U(x(t_0),y(t_0),z(t_0))- U(x(t_1),y(t_1),z(t_1))[/itex] is valid in any frame where [itex]\vec{F}(x,y,z) = -\nabla U(x,y,z)[/itex] where [itex]U(x,y,z)[/itex] does not vary with time in this frame.